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Realistic neuromorphic models and their application to neural reorganization simulations
Neurocomputing 48 (2002) 555–571
www.elsevier.com/locate/neucom
Realistic neuromorphic models and their application to neural reorganization simulations Regina C#elia Coelhoa; b , Luciano da Fontoura Costaa; ∗ a Cybernetic
Vision Research Group, Instituto de Fsica de S˜ao Carlos, Universidade de S˜ao Paulo, Caixa Postal 369, S˜ao Carlos 13560-970, SP, Brazil b Departamento de Inform atica, Centro de Tecnologia, Universidade Estadual de Maringa Campus Universitario, Av. Colombro, 5790, Maringa 87020-900, PR, Brazil Received 12 December 2000; accepted 25 May 2001
Abstract This article presents a comprehensive approach to the generation of morphologically realistic 2D and 3D neural cells. Statistical descriptions of key morphological parameters (such as dendritic lengths, widths and branching angles) are extracted from biological cells by using image analysis techniques and used to de4ne probability density functions, conditional or not, which are subsequently used to produce neural structures through Monte Carlo simulation and a graphic language. An application of the reported approach to the simulation of neurogenesis and neural reorganization processes considering attractive and repulsive c 2002 Elsevier Science B.V. All rights reserved. interaction 4elds is also included. Keywords: Neuromorphology; Neural simulation modeling; Neural reorganization
1. Introduction Modeling of neurons and neural structures has provided a powerful tool to examine and better understand the nervous system, especially since experiments with animals are often di9cult. Modeling has become particularly important because several parameters of biological neural structures—such as the connection patterns, neuron distribution, and the synaptic weights—are not known to the investigator. The situation is completely di;erent in a mathematic-computational model, where ∗ Corresponding author. Tel.: +55-16-273-9858; fax: +55-16-273-9879. E-mail addresses: [email protected], [email protected] (R. C#elia Coelho), [email protected] (L. da Fontoura Costa).
c 2002 Elsevier Science B.V. All rights reserved. 0925-2312/02/$ - see front matter PII: S 0 9 2 5 - 2 3 1 2 ( 0 1 ) 0 0 6 2 8 - 2
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all the parameters are not only known, but can also be easily changed. Thus, the development of realistic neuromorphic models, i.e. models of neural cells taking into account not only the biochemical parameters but also the morphology of the neurons, are essential for obtaining Dexible models and simulations and allowing the modeler to have full control of the several morphological and biochemical parameters involved. This is precisely the objective of the present article. More speci4cally, we address the problem of how to obtain synthetic neural cells and structures that are statistically similar to speci4c classes of neuronal cells found in nature. This has been done by deriving statistical models of relevant geometric properties of the considered neuronal cells, which are then used to produce the respective computational structures. Special attention is also given to the issue of neurogenesis, in order to allow the modeling of growing and reorganization processes. Many studies related to neural synthesis have shown that neuromorphology is an important factor in the behavior of the neural cell [18,22,29,38]. Speci4c examples include the several works relating functional and morphological classes, like in cat ganglion cells [2,11,30,31,33]. Other works have related the neural shapes with their receptive 4elds [7,21,37,38]. However, many works related to the neural simulation have used quite simple shapes, only taking into account characteristics such as the length and diameter of the dendritic and axonal segments, i.e., they do not include factors such as orientation or relative position. Electric-di;erential models to neural modeling such as NEURON [13] and GENESIS [3] or those based on the transmission cable analogy [3,15,28] often completely disregard several important morphological features, such as the branching angles between dendrites. One of the most important aspects related to morphologically realistic models is the attempt to relate how shape can constrain neural function, and vice versa. Branching patterns in most cells are very complex and their shapes present a high degree of variation. This paper proposes an approach to modeling and simulating morphologically realistic arti4cial neural structures through the use of statistics functions expressing several parameters, extracted by using image analysis techniques, describing the dendritic morphology. The dendritic trees are generated by random sampling (obeying uniform random functions) of these functions. The dendritic branches are generated by Monte-Carlo simulation over statistical models characterizing the neural morphometry, in such a way that resulting dendrites statistically match the shapes of the considered real neurons. It is shown that the used statistical approach is enough to generate neurons with realistic morphology. Such single cells are then combined to generate neural structures where several important parameters and e;ects governing cell growth can be carefully and completely controlled. The applications of morphologically realistic neural models are several. In the present work, one of them is presented, related to neurogenesis and neural reorganization processes. More speci4cally, it is investigated as to how the neural processes can be reorganized after depletion of neighboring cells, paying special attention to several types of 4eld interactions which may govern such a process.
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The article starts by presenting the methodology for generating single cells and neural structures, and follows by describing the neural reorganization application. 2. Methods 2.1. Generating single cells The possibility of using statistical techniques for neural shape generation has been outlined in some preliminary works [6,12,24]. Here probability density functions describing the principal shape features of the cells are used. Firstly, it is necessary to obtain representative shape measures to be used in the generation of the arti4cial cells. Such measures should be extracted from natural cells in order to produce realistic morphological models. This work has concentrated on cat ganglion cells, speci4cally alpha and beta cells. Measures extracted of 50 ganglion cells were used, including 23 alpha and 27 beta cells from several papers [2,8,11,16,21,31,35,36,39]. Only cells with eccentricity smaller than 3 mm have been considered in order to limit the otherwise substantial changes implied by varying eccentricity. These cells were originally stained by using di;erent methods. All those used by Boycott and WMassle, for instance, were stained using the Golgi–Cox method; those used by Fukuda and Leventhal were stained using HRP infections; and those from Saito used Lucifer yellow CH. The scanned neurons were then traced by using software developed by Lu#Ns Augusto Consularo of the Cybernetic Vision Research Group, 1 with the result being stored according to the Eutectic format. 2 This software was also used to calculate the measures considered during the generation of the arti4cial cells, which include: • number of primary branches, i.e., how many branches leave the soma; • length of the dendritic segments between an initial point (at the soma) and a
bifurcation point, between bifurcation points, or between a bifurcation point and an extremity; • angles between each segment and the previous segment; • diameter of each segment. These measures can be better understood by treating the neuron as a graph, as shown in Fig. 1. By adopting the Eutectic terminology, we have branch points (BP), natural ends (NE), middle tree origins (MTO), and continuous points (CP). The distance between a CP and the next or previous node will indicate the length of each straight line segment. The distance between a BP and the next BP or NE, or between MTO and the next BP will indicate the length of each dendritic segment. Two other measures calculated are the angles of straight line and angles of the dendritic segment. The diameter of each segment, as well as the probability 1 Cybernetic Vision Research Group is located in Insitute of Physics at S˜ ao Carlos, University of S˜ao Paulo, S˜ao Carlos, SP, Brazil. 2 Eutectic is a commercial format used for neural cell representation and storage.
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Fig. 1. Representation of part of a neuron in terms of a graph, considering the Eutectic terminology. Each arc represents a dendritic segment or a segment portion, and the nodes stand for branching points, tips, or high-curvature points.
of the rami4cation have also been considered. The last measure was the number of primary branches. All the measures, except the number of primary branches, are organized according to the subtree level, i.e. the hierarchical level along the tree, starting at the soma. For example, a primary dendritic segment (segment between an MTO and the BP) is a level one segment because it is on the level one subtree; one dendritic segment between the 4rst and second BP is on the level two subtree, therefore being a level two segment; and so on. The branch lengths are determined by adding the length of the straight line segments into each branch. By using these measures it is possible to estimate the probability density functions characterizing the morphological properties of the cells. It is from these functions that the distribution functions (DFs), also called cumulative distribution functions [14], are generated. A DF was estimated for each considered measure, generating a bivariated distribution function whose random components correspond to the hierarchical level and the respective measure (e.g. the angles, length, etc.). The angle values were always measured with respect to the previous segment. Examples of DFs are presented in Fig. 2. Fig. 2(a) and (b) present curves describing the branch length for 10 branching levels, respectively, to alpha and beta ganglion cells. It can be veri4ed that the curves are di;erent, especially with regard to the segment lengths. Fig. 2(c) and (d) present the diameter conditioned to segment length for alpha and beta ganglion cells, respectively. Firstly, it can be veri4ed that the segment lengths and diameters related to alpha cells are almost twice as large as beta cells. This indicates that alpha cells tend to be larger and thicker than beta cells. Besides, it can be noted that the relation between diameter and segment length is di;erent for alpha and beta cells. DFs characterizing other morphological parameters also turned out to be substantially di;erent for the two types of cells. For example, DFs relative to primary branch showed that beta cells have more primary branches than
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Fig. 2. Examples of DF curves of natural cells: curves for the branch length in the 10 levels in alpha and beta cells, respectively, (a) and (b); and the DF of the diameter conditional to the segment length to alpha and beta cells, respectively, (c) and (d). Observe that, as expected, the DFs tend to one as the observed random variable increases.
alpha cells. These di;erences are particularly important for the characterization of the cells. After the distribution functions are obtained, the neurons are generated by using Monte-Carlo simulation, and all the dendritic segments are generated in parallel. Therefore, the Monte-Carlo simulation involves randomly selecting a hierarchical level, drawing a random number between 0 and 1, and taking the corresponding abscissa as an instance of the random variable. Each segment is generated in terms of a series of smaller segments, in order to allow expression of the tortuosity at each branch. Still more realistic cells can be obtained by considering statistical descriptions of the adopted shape features conditioned to the branching order, that is, the generation of the current branch takes into account parameters at the previous stages. Conditional statistics have been considered in the examples as far as the diameter is concerned, in order to ensure that the diameters will always be reduced along the generation stages.
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Fig. 3. Examples of DF curves of arti4cial cells: curves for the branch length in the 10 levels in alpha and beta cells, respectively, (a) and (b); and the DF of the diameter conditional to the segment length to alpha and beta cells, respectively, (c) and (d). Observe that, as expected, the DFs tend to one as the observed random variable increases.
The comparison between real and arti4cial neural structures can be achieved by analyzing the DFs of the cells. Fig. 3 presents the same DFs as in Fig. 2, but now referring to the arti4cial cells. It can be veri4ed that the DFs are very similar. This indicates that the obtained arti4cial cells are morphologically similar to real ones. Three-dimensional cells have also been addressed, including all the measures considered in the two-dimensional case, but now two angles are needed to describe each straight-line segment and rami4cations. The curvature and torsion concepts from di;erential geometry [19] have been considered in order to de4ne these angles. Consider two straight lines R1 and R2 . In each case the deviation of R2 in relation to R1 (analogous to curvature) and the deviation of R2 from a plane (osculating plane) in a neighborhood of R1 (analogous to torsion) are measured. In Fig. 4 an example of calculation of the “curvature” and “torsion” angles can be seen. As shown in Fig. 4(a), the curvature angle () is the smallest angle between the R1 and R2 lines. The torsion (angle ’) indicates the extent to which
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Fig. 4. Example of geometric construction used to characterize the branching patterns in 3D cells: angle between the current and previous segments, approximating the curvature (a); and the angle ’ between the current and previous planes, approximating the torsion (b).
the plane 1 has to be rotated around R1 to reach plane 2 (Fig. 4(b)), 1 being the plane formed by R0 and R1 vectors, and 2 the plane formed by R1 and R2 vectors. 2.2. Generating neural structures Once having simulated neural growth individually, neural structures can be obtained through the combination of several neurons generated as previously described and including some biological factors during the growth and formation of structures. The main emphasis is put on the spatial development and the interaction of several neurons. Some important factors (such as the connections established during the growth of neurites, the random distribution of neurons in the structure, the existence of growth cones at the extremity of the axons, etc.) have to be considered in the generation of these structures, so that the result is realistic. All the branches and neurons grow simultaneously (in parallel). Each neuron possesses an axon and several dendrites. A connection is established whenever an axon approaches a dendrite (or vice versa), provided the dendrite and the axon belong to di;erent neurons. The synaptic strengths can be determined according to several models, e.g., in terms of the arc length from the synaptic contact to the axon hillock, a model which considers the electrotonic decay along the dendrites [1]. Another important aspect to be considered during the evolution of neural structures is the incorporation of growth cones mechanisms and neurogenesis models. Growth cones, which are situated at the extremity of axons, have the function of guiding the axons during their growth. Generally, they travel large distances before synapsing [4,17,20,32,34], being directed by many external inDuences during their journey.
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3. Results 3.1. Obtaining arti@cial neural structures Figs. 5 and 6 show examples of morphologically realistic 2D and 3D neural cells, respectively, obtained by using the method described in Section 2.1. Fig. 7 illustrates two 3D neurons visualized in terms of volumetric data techniques, and Fig. 8 illustrates some growth stages of a two-dimensional structure evolving in parallel. In real life, when a branch 4nds another on its way and no connection is made, it either stops growing or it changes its growth direction. In this example, we consider that the branches stop their growth whenever there is another on this way. The position of the cellular body of each cell is de4ned according to a uniform random distribution throughout the image space. In order to simulate the behavior of these structures in the axon, it has been considered that all neurites close (within a speci4c radius) to the axon will inDuence its evolution. The closer a neurite is to the axon extremity, the larger the inDuence will be. Thus, the axon tends to grow in the direction of larger neurite concentration, implying larger probability of synaptic connections. Fig. 9 illustrates this type of growth model.
Fig. 5. Examples of two-dimensional arti4cial cells generated from measures obtained from cat ganglion cells: -type cells (a); and -type cells (b). Observe that, as expected, the latter are smaller and more complex.
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Fig. 6. Some examples of three-dimensional arti4cial cells generated considering measures obtained from salamander ganglion cells: type one cells (a); and type two cells (b). Observe that type 1 cells are more complex.
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Fig. 7. Example of three-dimensional cells after application of volumetric data visualization techniques. Observe that the cell segments were dilated for the sake of better visualization.
Fig. 8. Stages along a simulated neurogenesis process involving -cells and allowing the segments to cross each other.
One important application illustrating the above-outlined possibilities, namely the simulation of neurogenesis processes, is described in the following section. 3.2. Simulation of neurogenesis processes During the formation of the mammalian nervous system, an excess of neural cells (about two or three times beyond what is necessary) is produced. Such
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Fig. 9. Example of a synthetic neural structure involving and ganglion cells. While the individual cell geometry is determined by the statistical characterization of the geometry of the respective natural cells, the axons are guided by the 4elds produced by the other cells. Every time an axon encounters a dendrite, a synaptic connection is established.
a process occurs, in general, during the initial days of life of the animal. Although not completely known, one of the main causes of neural death has been pinpointed to synaptic competition, which seems to favor survival of the neurons capable of establishing larger number of connections [25]. This natural death is essential for correct functional and structural maturation of the nervous system, since it can regulate the total number of neurons and to establish the correct pattern connections [5,9,27]. It has been veri4ed that after neural death processes in a young animal, a reorganization of the cells occurs. Perry and Linden [25,26] showed in their experiments that there is a reorientation of the dendrites of the ganglion cells of the Lister rats retina towards the depleted region. This implies that during development ganglion cell dendrites can compete for their territory and this competition can be important in the formation of the retina, like the uniform coverage by the dendritic cells and the regulation of cell survival [25,23]. The geometry of the dendrites of these cells also can be
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Fig. 10. Neural reorganization simulation under the inDuence of an attraction 4eld: (a) a neural structure with six 3D neurons; (b) reorganized structure with the dendritic arborization reoriented by the attraction 4eld produced by the two straight lines; (c) the dendritic arborization reoriented by the attraction 4eld produced by an ellipse; and (d) the same as before but considering a repulsive 4eld.
a;ected by the inDuence of neighboring ganglion cells during development [10,25]. Since the mechanisms governing such reorganization processes are not well known, modeling and simulating approaches can provide valuable insight into their underlying mechanisms. Here, the interest is focused on the investigation of the e;ect of several interaction 4elds governing the reorganization of the surviving neural cells. Fig. 10 illustrates the reorganization of a few 3D neural cells under two di;erent external inDuence 4elds de4ned by two orthogonal lines and an ellipse. The attraction 4elds follow the equation Fattract = kfL ;
(1)
where k is a constant that controls the speed in that the dendritic arborization will address to the attraction 4eld; f is calculated as a function 1=d2 with d being the distance between the point of neuron and each attraction=repulsion point; L the arc length from the current point to the soma; the constant that controls the force intensity.
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Fig. 11. Examples of neural reorganization for a two-dimensional tissue: (a) the original neural tissue; (b) the tissue reorganized under the inDuence of the attraction 4eld de4ned by the involving ellipse; (c) the same tissue reorganized according to a repulsive 4eld; (d) tissue in (a) with some central neurons removed; (e) the tissue reorganized under the inDuence of a Gaussian attraction 4eld at the center of the tissue.
The resulting structure for the 4eld de4ned by the straight lines is shown in Fig. 10(b), considering k = − 1=25; = 1:5. The reorganized structures under the inDuence of the ellipse-generated 4eld are presented in Figs. 10(c) and (d), respectively (k = − 1=15; = 1:5) and (k = − 1=10; = 1:0). Fig. 11 illustrates a simulation of neural reorganization for a two-dimensional tissue. Fig. 11(a) presents the original tissue; Fig. 11(b) presents the same tissue reorganized in the presence of an attraction 4eld (generated by the ellipse); and Fig. 11(c) shows the tissue reorganized due to the presence of a repulsion 4eld (also for an ellipse). Fig. 11(d) presents the depleted version of the tissue in (a) and Fig. 11(e) shows this tissue reorganized under a Gaussian attraction 4eld centered on the structure.
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4. Discussion The reported approach has opened new possibilities for modeling and simulating morphologically more realistic neural networks. It has become clear that one of the most interesting ways for synthesizing neural cells consists in considering distribution probability functions, conditional or not, capable of statistically describing the main morphological features of the simulated cells. By extracting a suitable set of shape measures of natural cells and by using them to estimate distribution functions and Monte-Carlo simulation, arti4cial cells presenting morphological characteristics close to those of natural cells can be obtained. This is particularly important since it allows a new approach to investigate the relationship between neural shape and function, with the additional advantage that all the morphological information about the produced structures is accurately known. By including a representation of the neural processes in terms of di;erential geometry concepts (analogous to torsion and curvature), the synthesis approach can be extended to produce 3D neural cells. Two- and three-dimensional neural tissues have been generated by using these arti4cial cells, which can incorporate several spatial distributions of the somata as well as the consideration of di;erent types of cells in the same tissue and neurotrophic inDuences. The synaptic connections, which occurred during the structure formation process, are established in terms of the proximity between axons and dendrites. Di;erent neural structures can be obtained by assuming distinct inDuence schemes, such as having dendritic extensions inversely proportional to the distance to the center of the image. In this way, several situations can be simulated in a more realistic way. A speci4c possibility would be to grow retinal ganglion cells with di;erentiated sizes, depending on their eccentricity. Several di;erent laws governing axon growth can be used in order to simulate di;erent attraction 4elds, and even include multiple axons. Also illustrated in the article was the application of the proposed methodology to simulations of neural reorganization processes. By considering several models governing reorganization, especially di;erent con4gurations of interaction 4elds, it is possible to get new insights into the biological processes controlling neural reorganization. Also, it is possible to simulate how the absence of the neighboring cells inDuences the development of the neurons around the depleted area. It should be observed that this reorganization depends of the animal age, since it is only observed in young animals. The younger the animal the more evident the reorganization [26]. Some possible factors that can be related with this reorganization have been presented by Perry and Ma;ei in the above paper. One of them can be related to a mutual inhibition of dendritic growth exerted by neighboring cells, in such a way that the absence of cells on one side would permit preferential growth in that direction. Another possibility is a competition for presynaptic contacts by dendrites of neighboring cells, where the dendrites that arrive later would have less chance to establish synaptic contacts, in such a way that the cells would grow preferentially into the depleted region. Chemotropic factors, according to [26], can also lead to this type of neural reorganization.
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Future developments should concentrate on the modeling of the inhibitory and=or excitatory gradient 4eld, like a tropism, where the external force slightly changes the direction of growth towards the center of the tropistic 4eld.
Acknowledgements Regina C#elia Coelho is grateful to FundaSca˜ o de Amparo aT Pesquisa do Estado de S˜ao Paulo (FAPESP-Brazil Proc. 98=05459-0) for 4nancial help. Luciano da F. Costa is grateful to FAPESP (Procs. 96=05497-3 and 94=04691-5) and Conselho Nacional de Desenvolvimento Cient#N4co e Tecnol#ogico (CNPq-Brazil Proc. 301422=92-3). The authors are indebted to Dr. Maria Cristina Ferreira de Oliveira and Dr. Rosane Minghim for the collaboration on the visualization of the threedimensional neurons. The authors are also grateful to Dr Toby J. Velte and Prof. R.J. Miller (Minnesota University) for supplying the salamander eutectic data from which some of the measures in this paper were extracted. The authors are also grateful to reviewers for comments and suggestions. References [1] J.A. Anderson, An Introduction to Neural Networks, MIT Press, Cambridge, 1995. [2] B.B. Boycott, H. WMassle, The morphological types of ganglion cells of the domestic cat’s retina, J. Physiol. 240 (1974) 397–419. [3] J.M. Bower, D. Beeman, The Book of GENESIS, Springer, New York, 1994. [4] J.F. Challacombe, D.M. Snow, P.C. Letourneau, Role of the cytoskeleton in growth cone motility and axonal elongation, Sem. Neurosci. 8 (1996) 67–80. [5] P.G.H. Clarke, Neuronal death in the development of the vertebrate nervous system, Trends Neurosci. Rev. (1985) 345–349. [6] L.F. Costa, R.M. Cesar Jr., R.C. Coelho, J.S. Tanaka, Analysis and synthesis of morphologically realistic neural networks, in: R. Poznanski (Ed.), Modelling in the Neurosciences: From Ionic Channels to Neural Networks, Harwood Academic Publishers, India, 1998, pp. 505–528. [7] D.M. Dacey, M.R. Petersen, Dendritic 4eld size and morphology of midget and parasol ganglion cells of the human retina, Proc. Natl. Acad. Sci. USA 89 (1992) 9666–9670. [8] J.F. Dann, E.H. Buhl, L. Peichl, Postnatal dendritic maturation of alpha and beta ganglion cells in cat retina, J. Neurosci. 8 (1988) 1485–1499. [9] J.E. Dowling, Neurons and Networks: An Introduction to Neuroscience, The Belhrap Press of Harvard University Press, Cambridge, 1992. [10] U.T. Eysel, L. Peichl, H. WMassle, Dendritic plasticity in the early feline retina: quantitative characteristics and sensitive period, J. Comp. Neurol. 242 (1985) 134–145. [11] Y. Fukuda, C.F. Hsiao, M. Watanabe, H. Ito, Morphological correlates of physiologically identi4ed Y-, X-, and W-cell in cat retina, J. Neurophysiol. 52 (1984) 999–1013. [12] P. Hamilton, A language to describe the growth of neurites, Biol. Cybern. 68 (1993) 559–565. [13] M.L. Hines, N.T. Carnevale, The NEURON simulation environment, Neural Comput. 9 (1997) 1179–1209. [14] E. Kreysziz, Advanced Engineering Mathematics, 7th Edition, Wiley, New York, USA, 1993.
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Luciano da Fontoura Costa was born in 1962 in S˜ao Carlos, SP, Brazil. He received a B.Sc. degree in Electronic Engineering from the University of S˜ao Paulo and a B.Sc. degree in Computer Science from the Federal University at S˜ao Carlos (Brazil), and got an M.Sc. degree in Applied Physics (Insitute of Physics at S˜ao Carlos—University of S˜ao Paulo) and a Ph.D. in Electronic Engineering from King’s College, University of London. He is member of the editorial boards of several journals, including Applied Signal Processing, Journal of Real-Time Imaging, and Psyche, and has acted as reviewer and invited editor for several journals. He has translated into Portuguese the book “Digital Image Processing” (by R. Gonzalez and R. Woods) and has written the book “Shape Analysis and Classi4cation: Theory and Practice” (CRC Press, 2000), both in collaboration with Dr. R.M. Cesar Jr. He has supervised more than 15 graduate programmes and written over 100 articles in international journals and conferences (http:==cyvision.if.sc.usp.br= ∼luciano=). Regina C!elia Coelho received a B.Sc. degree in Computer Science from State University Julio de Mesquita Filho—UNESP, Brazil, in 1989, an M.Sc. degree in Computer Science from Federal University at S˜ao Carlos, Brazil, in 1993, and a Ph.D. degree in Computational Physics from the Institute of Physics, University of S˜ao Paulo, Brazil, in 1998. She held a post-doctoral position at the Cybernetic Vision Research Group, Institute of Physics, University of S˜ao Paulo, 2000. She is currently a lecturer at State University of Maring#a. Her research interests include computational vision, image processing, computer graphic, biological vision, arti4cial intelligence, neuroscience, neural modeling and simulation.
www.elsevier.com/locate/neucom
Realistic neuromorphic models and their application to neural reorganization simulations Regina C#elia Coelhoa; b , Luciano da Fontoura Costaa; ∗ a Cybernetic
Vision Research Group, Instituto de Fsica de S˜ao Carlos, Universidade de S˜ao Paulo, Caixa Postal 369, S˜ao Carlos 13560-970, SP, Brazil b Departamento de Inform atica, Centro de Tecnologia, Universidade Estadual de Maringa Campus Universitario, Av. Colombro, 5790, Maringa 87020-900, PR, Brazil Received 12 December 2000; accepted 25 May 2001
Abstract This article presents a comprehensive approach to the generation of morphologically realistic 2D and 3D neural cells. Statistical descriptions of key morphological parameters (such as dendritic lengths, widths and branching angles) are extracted from biological cells by using image analysis techniques and used to de4ne probability density functions, conditional or not, which are subsequently used to produce neural structures through Monte Carlo simulation and a graphic language. An application of the reported approach to the simulation of neurogenesis and neural reorganization processes considering attractive and repulsive c 2002 Elsevier Science B.V. All rights reserved. interaction 4elds is also included. Keywords: Neuromorphology; Neural simulation modeling; Neural reorganization
1. Introduction Modeling of neurons and neural structures has provided a powerful tool to examine and better understand the nervous system, especially since experiments with animals are often di9cult. Modeling has become particularly important because several parameters of biological neural structures—such as the connection patterns, neuron distribution, and the synaptic weights—are not known to the investigator. The situation is completely di;erent in a mathematic-computational model, where ∗ Corresponding author. Tel.: +55-16-273-9858; fax: +55-16-273-9879. E-mail addresses: [email protected], [email protected] (R. C#elia Coelho), [email protected] (L. da Fontoura Costa).
c 2002 Elsevier Science B.V. All rights reserved. 0925-2312/02/$ - see front matter PII: S 0 9 2 5 - 2 3 1 2 ( 0 1 ) 0 0 6 2 8 - 2
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all the parameters are not only known, but can also be easily changed. Thus, the development of realistic neuromorphic models, i.e. models of neural cells taking into account not only the biochemical parameters but also the morphology of the neurons, are essential for obtaining Dexible models and simulations and allowing the modeler to have full control of the several morphological and biochemical parameters involved. This is precisely the objective of the present article. More speci4cally, we address the problem of how to obtain synthetic neural cells and structures that are statistically similar to speci4c classes of neuronal cells found in nature. This has been done by deriving statistical models of relevant geometric properties of the considered neuronal cells, which are then used to produce the respective computational structures. Special attention is also given to the issue of neurogenesis, in order to allow the modeling of growing and reorganization processes. Many studies related to neural synthesis have shown that neuromorphology is an important factor in the behavior of the neural cell [18,22,29,38]. Speci4c examples include the several works relating functional and morphological classes, like in cat ganglion cells [2,11,30,31,33]. Other works have related the neural shapes with their receptive 4elds [7,21,37,38]. However, many works related to the neural simulation have used quite simple shapes, only taking into account characteristics such as the length and diameter of the dendritic and axonal segments, i.e., they do not include factors such as orientation or relative position. Electric-di;erential models to neural modeling such as NEURON [13] and GENESIS [3] or those based on the transmission cable analogy [3,15,28] often completely disregard several important morphological features, such as the branching angles between dendrites. One of the most important aspects related to morphologically realistic models is the attempt to relate how shape can constrain neural function, and vice versa. Branching patterns in most cells are very complex and their shapes present a high degree of variation. This paper proposes an approach to modeling and simulating morphologically realistic arti4cial neural structures through the use of statistics functions expressing several parameters, extracted by using image analysis techniques, describing the dendritic morphology. The dendritic trees are generated by random sampling (obeying uniform random functions) of these functions. The dendritic branches are generated by Monte-Carlo simulation over statistical models characterizing the neural morphometry, in such a way that resulting dendrites statistically match the shapes of the considered real neurons. It is shown that the used statistical approach is enough to generate neurons with realistic morphology. Such single cells are then combined to generate neural structures where several important parameters and e;ects governing cell growth can be carefully and completely controlled. The applications of morphologically realistic neural models are several. In the present work, one of them is presented, related to neurogenesis and neural reorganization processes. More speci4cally, it is investigated as to how the neural processes can be reorganized after depletion of neighboring cells, paying special attention to several types of 4eld interactions which may govern such a process.
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The article starts by presenting the methodology for generating single cells and neural structures, and follows by describing the neural reorganization application. 2. Methods 2.1. Generating single cells The possibility of using statistical techniques for neural shape generation has been outlined in some preliminary works [6,12,24]. Here probability density functions describing the principal shape features of the cells are used. Firstly, it is necessary to obtain representative shape measures to be used in the generation of the arti4cial cells. Such measures should be extracted from natural cells in order to produce realistic morphological models. This work has concentrated on cat ganglion cells, speci4cally alpha and beta cells. Measures extracted of 50 ganglion cells were used, including 23 alpha and 27 beta cells from several papers [2,8,11,16,21,31,35,36,39]. Only cells with eccentricity smaller than 3 mm have been considered in order to limit the otherwise substantial changes implied by varying eccentricity. These cells were originally stained by using di;erent methods. All those used by Boycott and WMassle, for instance, were stained using the Golgi–Cox method; those used by Fukuda and Leventhal were stained using HRP infections; and those from Saito used Lucifer yellow CH. The scanned neurons were then traced by using software developed by Lu#Ns Augusto Consularo of the Cybernetic Vision Research Group, 1 with the result being stored according to the Eutectic format. 2 This software was also used to calculate the measures considered during the generation of the arti4cial cells, which include: • number of primary branches, i.e., how many branches leave the soma; • length of the dendritic segments between an initial point (at the soma) and a
bifurcation point, between bifurcation points, or between a bifurcation point and an extremity; • angles between each segment and the previous segment; • diameter of each segment. These measures can be better understood by treating the neuron as a graph, as shown in Fig. 1. By adopting the Eutectic terminology, we have branch points (BP), natural ends (NE), middle tree origins (MTO), and continuous points (CP). The distance between a CP and the next or previous node will indicate the length of each straight line segment. The distance between a BP and the next BP or NE, or between MTO and the next BP will indicate the length of each dendritic segment. Two other measures calculated are the angles of straight line and angles of the dendritic segment. The diameter of each segment, as well as the probability 1 Cybernetic Vision Research Group is located in Insitute of Physics at S˜ ao Carlos, University of S˜ao Paulo, S˜ao Carlos, SP, Brazil. 2 Eutectic is a commercial format used for neural cell representation and storage.
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Fig. 1. Representation of part of a neuron in terms of a graph, considering the Eutectic terminology. Each arc represents a dendritic segment or a segment portion, and the nodes stand for branching points, tips, or high-curvature points.
of the rami4cation have also been considered. The last measure was the number of primary branches. All the measures, except the number of primary branches, are organized according to the subtree level, i.e. the hierarchical level along the tree, starting at the soma. For example, a primary dendritic segment (segment between an MTO and the BP) is a level one segment because it is on the level one subtree; one dendritic segment between the 4rst and second BP is on the level two subtree, therefore being a level two segment; and so on. The branch lengths are determined by adding the length of the straight line segments into each branch. By using these measures it is possible to estimate the probability density functions characterizing the morphological properties of the cells. It is from these functions that the distribution functions (DFs), also called cumulative distribution functions [14], are generated. A DF was estimated for each considered measure, generating a bivariated distribution function whose random components correspond to the hierarchical level and the respective measure (e.g. the angles, length, etc.). The angle values were always measured with respect to the previous segment. Examples of DFs are presented in Fig. 2. Fig. 2(a) and (b) present curves describing the branch length for 10 branching levels, respectively, to alpha and beta ganglion cells. It can be veri4ed that the curves are di;erent, especially with regard to the segment lengths. Fig. 2(c) and (d) present the diameter conditioned to segment length for alpha and beta ganglion cells, respectively. Firstly, it can be veri4ed that the segment lengths and diameters related to alpha cells are almost twice as large as beta cells. This indicates that alpha cells tend to be larger and thicker than beta cells. Besides, it can be noted that the relation between diameter and segment length is di;erent for alpha and beta cells. DFs characterizing other morphological parameters also turned out to be substantially di;erent for the two types of cells. For example, DFs relative to primary branch showed that beta cells have more primary branches than
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Fig. 2. Examples of DF curves of natural cells: curves for the branch length in the 10 levels in alpha and beta cells, respectively, (a) and (b); and the DF of the diameter conditional to the segment length to alpha and beta cells, respectively, (c) and (d). Observe that, as expected, the DFs tend to one as the observed random variable increases.
alpha cells. These di;erences are particularly important for the characterization of the cells. After the distribution functions are obtained, the neurons are generated by using Monte-Carlo simulation, and all the dendritic segments are generated in parallel. Therefore, the Monte-Carlo simulation involves randomly selecting a hierarchical level, drawing a random number between 0 and 1, and taking the corresponding abscissa as an instance of the random variable. Each segment is generated in terms of a series of smaller segments, in order to allow expression of the tortuosity at each branch. Still more realistic cells can be obtained by considering statistical descriptions of the adopted shape features conditioned to the branching order, that is, the generation of the current branch takes into account parameters at the previous stages. Conditional statistics have been considered in the examples as far as the diameter is concerned, in order to ensure that the diameters will always be reduced along the generation stages.
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Fig. 3. Examples of DF curves of arti4cial cells: curves for the branch length in the 10 levels in alpha and beta cells, respectively, (a) and (b); and the DF of the diameter conditional to the segment length to alpha and beta cells, respectively, (c) and (d). Observe that, as expected, the DFs tend to one as the observed random variable increases.
The comparison between real and arti4cial neural structures can be achieved by analyzing the DFs of the cells. Fig. 3 presents the same DFs as in Fig. 2, but now referring to the arti4cial cells. It can be veri4ed that the DFs are very similar. This indicates that the obtained arti4cial cells are morphologically similar to real ones. Three-dimensional cells have also been addressed, including all the measures considered in the two-dimensional case, but now two angles are needed to describe each straight-line segment and rami4cations. The curvature and torsion concepts from di;erential geometry [19] have been considered in order to de4ne these angles. Consider two straight lines R1 and R2 . In each case the deviation of R2 in relation to R1 (analogous to curvature) and the deviation of R2 from a plane (osculating plane) in a neighborhood of R1 (analogous to torsion) are measured. In Fig. 4 an example of calculation of the “curvature” and “torsion” angles can be seen. As shown in Fig. 4(a), the curvature angle () is the smallest angle between the R1 and R2 lines. The torsion (angle ’) indicates the extent to which
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Fig. 4. Example of geometric construction used to characterize the branching patterns in 3D cells: angle between the current and previous segments, approximating the curvature (a); and the angle ’ between the current and previous planes, approximating the torsion (b).
the plane 1 has to be rotated around R1 to reach plane 2 (Fig. 4(b)), 1 being the plane formed by R0 and R1 vectors, and 2 the plane formed by R1 and R2 vectors. 2.2. Generating neural structures Once having simulated neural growth individually, neural structures can be obtained through the combination of several neurons generated as previously described and including some biological factors during the growth and formation of structures. The main emphasis is put on the spatial development and the interaction of several neurons. Some important factors (such as the connections established during the growth of neurites, the random distribution of neurons in the structure, the existence of growth cones at the extremity of the axons, etc.) have to be considered in the generation of these structures, so that the result is realistic. All the branches and neurons grow simultaneously (in parallel). Each neuron possesses an axon and several dendrites. A connection is established whenever an axon approaches a dendrite (or vice versa), provided the dendrite and the axon belong to di;erent neurons. The synaptic strengths can be determined according to several models, e.g., in terms of the arc length from the synaptic contact to the axon hillock, a model which considers the electrotonic decay along the dendrites [1]. Another important aspect to be considered during the evolution of neural structures is the incorporation of growth cones mechanisms and neurogenesis models. Growth cones, which are situated at the extremity of axons, have the function of guiding the axons during their growth. Generally, they travel large distances before synapsing [4,17,20,32,34], being directed by many external inDuences during their journey.
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3. Results 3.1. Obtaining arti@cial neural structures Figs. 5 and 6 show examples of morphologically realistic 2D and 3D neural cells, respectively, obtained by using the method described in Section 2.1. Fig. 7 illustrates two 3D neurons visualized in terms of volumetric data techniques, and Fig. 8 illustrates some growth stages of a two-dimensional structure evolving in parallel. In real life, when a branch 4nds another on its way and no connection is made, it either stops growing or it changes its growth direction. In this example, we consider that the branches stop their growth whenever there is another on this way. The position of the cellular body of each cell is de4ned according to a uniform random distribution throughout the image space. In order to simulate the behavior of these structures in the axon, it has been considered that all neurites close (within a speci4c radius) to the axon will inDuence its evolution. The closer a neurite is to the axon extremity, the larger the inDuence will be. Thus, the axon tends to grow in the direction of larger neurite concentration, implying larger probability of synaptic connections. Fig. 9 illustrates this type of growth model.
Fig. 5. Examples of two-dimensional arti4cial cells generated from measures obtained from cat ganglion cells: -type cells (a); and -type cells (b). Observe that, as expected, the latter are smaller and more complex.
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Fig. 6. Some examples of three-dimensional arti4cial cells generated considering measures obtained from salamander ganglion cells: type one cells (a); and type two cells (b). Observe that type 1 cells are more complex.
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Fig. 7. Example of three-dimensional cells after application of volumetric data visualization techniques. Observe that the cell segments were dilated for the sake of better visualization.
Fig. 8. Stages along a simulated neurogenesis process involving -cells and allowing the segments to cross each other.
One important application illustrating the above-outlined possibilities, namely the simulation of neurogenesis processes, is described in the following section. 3.2. Simulation of neurogenesis processes During the formation of the mammalian nervous system, an excess of neural cells (about two or three times beyond what is necessary) is produced. Such
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Fig. 9. Example of a synthetic neural structure involving and ganglion cells. While the individual cell geometry is determined by the statistical characterization of the geometry of the respective natural cells, the axons are guided by the 4elds produced by the other cells. Every time an axon encounters a dendrite, a synaptic connection is established.
a process occurs, in general, during the initial days of life of the animal. Although not completely known, one of the main causes of neural death has been pinpointed to synaptic competition, which seems to favor survival of the neurons capable of establishing larger number of connections [25]. This natural death is essential for correct functional and structural maturation of the nervous system, since it can regulate the total number of neurons and to establish the correct pattern connections [5,9,27]. It has been veri4ed that after neural death processes in a young animal, a reorganization of the cells occurs. Perry and Linden [25,26] showed in their experiments that there is a reorientation of the dendrites of the ganglion cells of the Lister rats retina towards the depleted region. This implies that during development ganglion cell dendrites can compete for their territory and this competition can be important in the formation of the retina, like the uniform coverage by the dendritic cells and the regulation of cell survival [25,23]. The geometry of the dendrites of these cells also can be
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Fig. 10. Neural reorganization simulation under the inDuence of an attraction 4eld: (a) a neural structure with six 3D neurons; (b) reorganized structure with the dendritic arborization reoriented by the attraction 4eld produced by the two straight lines; (c) the dendritic arborization reoriented by the attraction 4eld produced by an ellipse; and (d) the same as before but considering a repulsive 4eld.
a;ected by the inDuence of neighboring ganglion cells during development [10,25]. Since the mechanisms governing such reorganization processes are not well known, modeling and simulating approaches can provide valuable insight into their underlying mechanisms. Here, the interest is focused on the investigation of the e;ect of several interaction 4elds governing the reorganization of the surviving neural cells. Fig. 10 illustrates the reorganization of a few 3D neural cells under two di;erent external inDuence 4elds de4ned by two orthogonal lines and an ellipse. The attraction 4elds follow the equation Fattract = kfL ;
(1)
where k is a constant that controls the speed in that the dendritic arborization will address to the attraction 4eld; f is calculated as a function 1=d2 with d being the distance between the point of neuron and each attraction=repulsion point; L the arc length from the current point to the soma; the constant that controls the force intensity.
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Fig. 11. Examples of neural reorganization for a two-dimensional tissue: (a) the original neural tissue; (b) the tissue reorganized under the inDuence of the attraction 4eld de4ned by the involving ellipse; (c) the same tissue reorganized according to a repulsive 4eld; (d) tissue in (a) with some central neurons removed; (e) the tissue reorganized under the inDuence of a Gaussian attraction 4eld at the center of the tissue.
The resulting structure for the 4eld de4ned by the straight lines is shown in Fig. 10(b), considering k = − 1=25; = 1:5. The reorganized structures under the inDuence of the ellipse-generated 4eld are presented in Figs. 10(c) and (d), respectively (k = − 1=15; = 1:5) and (k = − 1=10; = 1:0). Fig. 11 illustrates a simulation of neural reorganization for a two-dimensional tissue. Fig. 11(a) presents the original tissue; Fig. 11(b) presents the same tissue reorganized in the presence of an attraction 4eld (generated by the ellipse); and Fig. 11(c) shows the tissue reorganized due to the presence of a repulsion 4eld (also for an ellipse). Fig. 11(d) presents the depleted version of the tissue in (a) and Fig. 11(e) shows this tissue reorganized under a Gaussian attraction 4eld centered on the structure.
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4. Discussion The reported approach has opened new possibilities for modeling and simulating morphologically more realistic neural networks. It has become clear that one of the most interesting ways for synthesizing neural cells consists in considering distribution probability functions, conditional or not, capable of statistically describing the main morphological features of the simulated cells. By extracting a suitable set of shape measures of natural cells and by using them to estimate distribution functions and Monte-Carlo simulation, arti4cial cells presenting morphological characteristics close to those of natural cells can be obtained. This is particularly important since it allows a new approach to investigate the relationship between neural shape and function, with the additional advantage that all the morphological information about the produced structures is accurately known. By including a representation of the neural processes in terms of di;erential geometry concepts (analogous to torsion and curvature), the synthesis approach can be extended to produce 3D neural cells. Two- and three-dimensional neural tissues have been generated by using these arti4cial cells, which can incorporate several spatial distributions of the somata as well as the consideration of di;erent types of cells in the same tissue and neurotrophic inDuences. The synaptic connections, which occurred during the structure formation process, are established in terms of the proximity between axons and dendrites. Di;erent neural structures can be obtained by assuming distinct inDuence schemes, such as having dendritic extensions inversely proportional to the distance to the center of the image. In this way, several situations can be simulated in a more realistic way. A speci4c possibility would be to grow retinal ganglion cells with di;erentiated sizes, depending on their eccentricity. Several di;erent laws governing axon growth can be used in order to simulate di;erent attraction 4elds, and even include multiple axons. Also illustrated in the article was the application of the proposed methodology to simulations of neural reorganization processes. By considering several models governing reorganization, especially di;erent con4gurations of interaction 4elds, it is possible to get new insights into the biological processes controlling neural reorganization. Also, it is possible to simulate how the absence of the neighboring cells inDuences the development of the neurons around the depleted area. It should be observed that this reorganization depends of the animal age, since it is only observed in young animals. The younger the animal the more evident the reorganization [26]. Some possible factors that can be related with this reorganization have been presented by Perry and Ma;ei in the above paper. One of them can be related to a mutual inhibition of dendritic growth exerted by neighboring cells, in such a way that the absence of cells on one side would permit preferential growth in that direction. Another possibility is a competition for presynaptic contacts by dendrites of neighboring cells, where the dendrites that arrive later would have less chance to establish synaptic contacts, in such a way that the cells would grow preferentially into the depleted region. Chemotropic factors, according to [26], can also lead to this type of neural reorganization.
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Future developments should concentrate on the modeling of the inhibitory and=or excitatory gradient 4eld, like a tropism, where the external force slightly changes the direction of growth towards the center of the tropistic 4eld.
Acknowledgements Regina C#elia Coelho is grateful to FundaSca˜ o de Amparo aT Pesquisa do Estado de S˜ao Paulo (FAPESP-Brazil Proc. 98=05459-0) for 4nancial help. Luciano da F. Costa is grateful to FAPESP (Procs. 96=05497-3 and 94=04691-5) and Conselho Nacional de Desenvolvimento Cient#N4co e Tecnol#ogico (CNPq-Brazil Proc. 301422=92-3). The authors are indebted to Dr. Maria Cristina Ferreira de Oliveira and Dr. Rosane Minghim for the collaboration on the visualization of the threedimensional neurons. The authors are also grateful to Dr Toby J. Velte and Prof. R.J. Miller (Minnesota University) for supplying the salamander eutectic data from which some of the measures in this paper were extracted. The authors are also grateful to reviewers for comments and suggestions. References [1] J.A. Anderson, An Introduction to Neural Networks, MIT Press, Cambridge, 1995. [2] B.B. Boycott, H. WMassle, The morphological types of ganglion cells of the domestic cat’s retina, J. Physiol. 240 (1974) 397–419. [3] J.M. Bower, D. Beeman, The Book of GENESIS, Springer, New York, 1994. [4] J.F. Challacombe, D.M. Snow, P.C. Letourneau, Role of the cytoskeleton in growth cone motility and axonal elongation, Sem. Neurosci. 8 (1996) 67–80. [5] P.G.H. Clarke, Neuronal death in the development of the vertebrate nervous system, Trends Neurosci. Rev. (1985) 345–349. [6] L.F. Costa, R.M. Cesar Jr., R.C. Coelho, J.S. Tanaka, Analysis and synthesis of morphologically realistic neural networks, in: R. Poznanski (Ed.), Modelling in the Neurosciences: From Ionic Channels to Neural Networks, Harwood Academic Publishers, India, 1998, pp. 505–528. [7] D.M. Dacey, M.R. Petersen, Dendritic 4eld size and morphology of midget and parasol ganglion cells of the human retina, Proc. Natl. Acad. Sci. USA 89 (1992) 9666–9670. [8] J.F. Dann, E.H. Buhl, L. Peichl, Postnatal dendritic maturation of alpha and beta ganglion cells in cat retina, J. Neurosci. 8 (1988) 1485–1499. [9] J.E. Dowling, Neurons and Networks: An Introduction to Neuroscience, The Belhrap Press of Harvard University Press, Cambridge, 1992. [10] U.T. Eysel, L. Peichl, H. WMassle, Dendritic plasticity in the early feline retina: quantitative characteristics and sensitive period, J. Comp. Neurol. 242 (1985) 134–145. [11] Y. Fukuda, C.F. Hsiao, M. Watanabe, H. Ito, Morphological correlates of physiologically identi4ed Y-, X-, and W-cell in cat retina, J. Neurophysiol. 52 (1984) 999–1013. [12] P. Hamilton, A language to describe the growth of neurites, Biol. Cybern. 68 (1993) 559–565. [13] M.L. Hines, N.T. Carnevale, The NEURON simulation environment, Neural Comput. 9 (1997) 1179–1209. [14] E. Kreysziz, Advanced Engineering Mathematics, 7th Edition, Wiley, New York, USA, 1993.
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Luciano da Fontoura Costa was born in 1962 in S˜ao Carlos, SP, Brazil. He received a B.Sc. degree in Electronic Engineering from the University of S˜ao Paulo and a B.Sc. degree in Computer Science from the Federal University at S˜ao Carlos (Brazil), and got an M.Sc. degree in Applied Physics (Insitute of Physics at S˜ao Carlos—University of S˜ao Paulo) and a Ph.D. in Electronic Engineering from King’s College, University of London. He is member of the editorial boards of several journals, including Applied Signal Processing, Journal of Real-Time Imaging, and Psyche, and has acted as reviewer and invited editor for several journals. He has translated into Portuguese the book “Digital Image Processing” (by R. Gonzalez and R. Woods) and has written the book “Shape Analysis and Classi4cation: Theory and Practice” (CRC Press, 2000), both in collaboration with Dr. R.M. Cesar Jr. He has supervised more than 15 graduate programmes and written over 100 articles in international journals and conferences (http:==cyvision.if.sc.usp.br= ∼luciano=). Regina C!elia Coelho received a B.Sc. degree in Computer Science from State University Julio de Mesquita Filho—UNESP, Brazil, in 1989, an M.Sc. degree in Computer Science from Federal University at S˜ao Carlos, Brazil, in 1993, and a Ph.D. degree in Computational Physics from the Institute of Physics, University of S˜ao Paulo, Brazil, in 1998. She held a post-doctoral position at the Cybernetic Vision Research Group, Institute of Physics, University of S˜ao Paulo, 2000. She is currently a lecturer at State University of Maring#a. Her research interests include computational vision, image processing, computer graphic, biological vision, arti4cial intelligence, neuroscience, neural modeling and simulation.