- Email: [email protected]
Geometric modeling of local cortical networks
Neurocomputing 32}33 (2000) 461}469
Geometric modeling of local cortical networks B.H. McCormick *, W. Koh , W.R. Shankle, J.H. Fallon Scientixc Visualization Laboratory, Department of Computer Science, Texas A&M University, College Station, TX 77843-3112, USA Department of Cognitive Sciences, University of California, Irvine, CA 92697, USA Department of Anatomy and Neurobiology, University of California, Irvine, CA 92697, USA Accepted 13 January 2000
Abstract Geometric models of local cortical microstructure can be used to develop neuroanatomically based network models. First, neuron/"ber spatial distributions and neuron morphology, drawing on a new microstructure imaging technology, must be quanti"ed within the cortical tissue. Given this database, stochastic models recreating the spatial distribution and morphology of the observed neurons are constructed. Synthetic neuron forests are then stochastically generated. Neurite near-collisions (e.g., between axons and dendrites) are modeled to add synaptic connections. The synapse estimation complements the forest of synthetic neurons by adding local connectivity, i.e., wiring up the forest into network models. 2000 Published by Elsevier Science B.V. All rights reserved. Keywords: Cortex; Neuroanatomy; Brain connectivity; Hierarchical cortical network models; Stochastic modeling of synapse formation
1. Introduction The connectivity of human and other mammalian brains is largely uncharted. In our time, large-scale telescopes have opened the external universe to measurement and modeling of its evolution. In the same way, new large-scale light microscopes, processing teravoxels of image data per day [8], promise to open the internal connectivity of brains of all species, including human, to measurement and modeling
* Corresponding author. Tel.: #1-979-845-8870; fax: #1-979-847-8578. E-mail address: [email protected] (B.H. McCormick). 0925-2312/00/$ - see front matter 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 2 3 1 2 ( 0 0 ) 0 0 2 0 0 - 9
462
B.H. McCormick et al. / Neurocomputing 32}33 (2000) 461}469
of brain development and evolution, a natural precursor to detailed functional modeling of the brain. Anticipating this technological development, we describe below a new methodology that recasts three-dimensional reconstruction of brain microstructure into models of local cortical networks. This will allow computational neuroscientists to incorporate neuroanatomical realism into their models. 1.1. Global connectivity databases and their analysis Enno-Stefan and others at the C&O Vogt Institute in Germany and at Newcastleupon-Tyne University in England have collated hundreds of studies on the connectivity of the macaque monkey cerebral cortex [15,14,11,6]. They have created a database that allows one to de"ne the known connectivity, their level of certainty and resolution, and relate them across di!erent cytoarchitectural classi"cation schemes. This invaluable resource represents a set of hypotheses for the connectivity of other primates, including Homo Sapiens. As such, they will be used to compare interspecies cortical connectivity maps as well as to "ll in missing information until human data become available. 1.2. Modeling local connections vs. long-range connections Modeling long-range connections (i.e., association "bers) will be much more di$cult than modeling short-range connections. As a general principal, the serial progression of cortico-cortical projections from primary to secondary to tertiary to higher-order association cortices is known. Nonetheless, at a "ner level of detail, the patterns of association "bers are highly speci"c, arranged in a myriad of parallel projections, and do not follow any obvious rules. The factors that lead some pairs of areas to form connections over considerable distances (e.g., from occipital to frontal cortex), whereas others do not, are not understood. Until they have been identi"ed, modeling (other than database modeling) of long-range connections may prove intractable. For this reason the geometric models of cortical microstructure proposed here focus on local connectivity.
2. Hierarchical network models of brain connectivity 2.1. What constitutes brain connectivity? The brain can be represented as a collection of interconnected neurons. Brain connectivity can be described at four levels of detail (LOD). Our approach is to carefully characterize brain connectivity at the level of volume-"lling circuitry, and describe pooling strategies to derive characterizations of brain connectivity at coarser levels of detail. The contribution of glial cells, blood vasculature, the ventricles, and the like will be ignored below.
B.H. McCormick et al. / Neurocomputing 32}33 (2000) 461}469
463
2.2. Levels of detail E Volume-xlling circuitry (exhibiting cell-to-cell connectivity) records which neuron (and type) is the source of the axon, where the axonal arbor projects (into which pools of neurons); and for each of its axonal segments, ascertains the type and location of all its synapses on postsynaptic neurons. E Population-based connectivity (from neuron morphology and "ber tract modeling). Population-based connectivity modeling aims to capture the local circuitry in a manner that may be generalized over a small population of source and target neurons and their mutual connections. E Finite element modeling of brain connectivity, where discrete bundles of "bers reach across the brain to link the hexahedral elements in a "nite element model of the brain. A typical Brodmann area in human cerebral cortex is modeled by a threedimensional mesh of 50 elements (4 mm;4 mm;3 mm), and similar numbers are needed for other brain nuclei. Finer grain may be required for the resolution of subnuclei. E Global connectivity at a nucleus/cortical area level (e.g., Fellman and Van Essen's hierarchical connection matrix for monkey visual cortex [3]; the C&O Vogt ORT Connectivity Database; G. Burns's connectivity database for the rat brain). 2.3. Brain tissue connectivity at a volume-xlling level of detail Connections are mediated through synapses [12]. A synapse is a connection between a presynaptic and a postsynaptic process (e.g., between a source axon and a target dendrite). Envision the following gedanken experiment: each synapse is cut and split into its pre- and post-synaptic terminals. These terminals are labeled with a universal synaptic address and then assigned, respectively, to their pre- and postsynaptic neurons. This information completely describes the neural connectivity of the tissue: its neural network can be reconstructed solely from a description of its individual neurons.
3. Web-based description of brain tissue The Web-based implementation of our distributed database for brain microstructure [7] includes an exploratory environment, an Intranet, in which to model the pooling strategies needed to characterize local cortical connectivity at coarser LOD than the ideal level, namely, volume-"lling circuitry. The interpretation is both natural and straightforward. Web pages are assigned to all "les, and in particular, to the synapse "les described in the distributed database system. The user, examining a given dendritic segment of the neuron, can open the synapse page for the segment and select a postsynaptic terminal. A mouse click links the user to the associated axonal segment page of the presynaptic neuron, and identi"es the associated presynaptic terminal. Similar strategies can be applied to axonal segments and soma.
464
B.H. McCormick et al. / Neurocomputing 32}33 (2000) 461}469
4. Construction of network models based on the neuroanatomy 4.1. Estimating synapses from neurite near-collisions In principle, the synaptology at a volume-xlling circuitry level of detail of all neurons within a "nite element can be estimated. How is this possible? Consider the following gedanken experiment: generate a synthetic forest of neurons "lling the speci"ed "nite element, where the forest preserves the spatial distributions (layer-by-layer) and neuron morphologies observed within the "nite element [10]. Then examining each synthetic neuron in turn, evaluate the probability that its soma (considered as a segment at level 0) and higher-level dendritic segments form potential synapses with neighboring "bers. These neighboring "bers, candidates for synapse formation, can be axonal segments of neighboring neurons, recurrent axon collaterals, or association "bers. For each "ber there is a point of closest approach by a dendritic segment, and it is at this point that a synapse is most likely to form (Fig. 1). To complete the process we need to invoke the `principle of the promiscuous neurona, to wit `if a dendrite and a "ber can form a more perfect union (a synapse), they willa. Clearly, the probability of a synaptic union drops o! sharply with the inter-segment distance r. But within 1}4 lm, the probability of spines developing and synapses forming is high [12]. Christine Harris, in her recent lapsed-time video of synapse formation in cultured neurons [5], shows budding spines `gropinga for one another. So, quite as growthbased modeling of neuron morphology is driven by knowledge of growth cone dynamics, the synaptology of brain tissue will be driven by knowledge of spine dynamics and the groping process by which synapses are formed. What we have shown above is that, in principle, we could "ll out in complete detail, including its synaptology, the distributed database of neuron morphology.
Fig. 1. Point of closet approach of dendritic segment to an axial "ber (axon) [9].
B.H. McCormick et al. / Neurocomputing 32}33 (2000) 461}469
465
Furthermore, the resulting distribution of synapses could be con"rmed by synapsebased staining. 4.2. Geometric models of local cortical networks What is required to make the above process computationally tractable? First, the synthetic neurons of the "nite element are partitioned into classes (population pools) of known probability. Neurons are spatially partitioned by the position of their soma into concentric stacks of wafers, each wafer modeling one cortical layer centered around a given vertical axis (Figs. 2 and 3). Then, the synthetic neurons within each wafer are partitioned by cell morphology into disjoint "ner-grained subclasses, from which prototypical neurons are selected. Next, the spatial distribution of "bers (both recurrent axon collaterals and association "bers) that can impinge upon the dendritic arbors of these prototypical neurons is similarly modeled. By this process the immense number of neurons (and axonal "bers) in the "nite element has now been reduced to a tractable number of prototypical neurons and "bers. Using these prototypes, the appropriate spatial distribution of dendritic segments and "bers is estimated by a standard Monte Carlo simulation. Though not universally applicable, translational and rotational invariance can be evoked to simplify the spatial distribution functions [9]. This procedure sketches the methodology to estimate the intrinsic connectivity in the developing human cerebral cortex. Models of the human cerebral cortex will be needed for all cortical areas, agepoints, and neurological diagnoses. These models, in fact, describe not only the
Fig. 2. Modeling environment for cylindrical module of cells [9].
466
B.H. McCormick et al. / Neurocomputing 32}33 (2000) 461}469
Fig. 3. Interaction at the wafer level [9].
population-based local connectivity of the tissue, but also describe canonical cortical models of the neural circuitry. To our knowledge, these methods are new and described here for the "rst time. Constructing anatomical-based neural microcircuits in this way}driven directly from measurements of the cortical microstructure represents a signi"cant advance in computational neuroscience.
4.3. Finite element modeling of brain connectivity Consider all neurons of a given type whose soma resides in a "nite element at a given locus in the cerebral cortex [2]. Imagine that by injecting a bolus of stain into the "nite element that the soma, dendritic arbors, and axonal arbors of all these cells, and only these cells, were completely "lled with stain. Changing from a neuronal perspective to a "nite element level of detail, we now pool the neurons of the "nite element together to derive a pooled description of brain connectivity. Speci"cally, we visualize the "nite element as the `somaa of a `fat neurona, which in general has discrete bundles of association "bers reaching out across the brain. What, at a neuronal level of detail, were synapses between an individual association "ber and the dendrites and soma of its target neurons become, at this coarser level of description, `fat synapsesa between the bundles of association "bers and the "nite elements housing the target neurons. This example shows that to describe brain connectivity at a "nite element level of detail (or, more generally, at any of the three lower levels of detail) we must establish a formal morphological description of these "nite element (population-based) pools. And replacing the synapse, we must establish conventions for what constitutes a `synapsea at this coarser level of description. In e!ect, the pooled neurons, considered as the sum of their parts, carry too much detailed information. Our task is to "nd what parts of this information can be thrown away without jeopardizing the description of brain connectivity at lower levels of detail.
B.H. McCormick et al. / Neurocomputing 32}33 (2000) 461}469
467
4.4. Hierarchical network structure and gauge theory Gross and Chen [4] have developed a systematic approach to manipulate hierarchical network structures. In this approach, large-scale networks with certain symmetry are represented in a very compact manner that greatly reduces the computational complexity. This approach "ts very well our characterization of brain connectivity described at the four levels of details, which form a hierarchy of networks. Moreover, we are investigating the relation between this representation and gauge theory in modern particle physics and condensed matter physics [13]. Gauge theory is a powerful mathematical tool in the study of such hierarchical networks.
5. Interface to multi-compartmental modeling of cortical microcircuits The overriding objective in these studies is to create canonical geometric models of local cortical microstructure, by cytoarchitectural area, age and neurological diagnosis, that can be used to develop neuroanatomical-based cortical circuit models. We propose to provide an interface to GENESIS [1], a widely used neural simulator, to facilitate this transition from geometric models of local cortical microstructure to dynamic modeling of the neuroanatomical-based cortical circuits.
Acknowledgements This work was supported in part by Texas Advanced Technology Program Grant 999903-124 (McCormick) from the Texas Higher Education Coordinating Board.
References [1] J. Bower, D. Beeman, The Book of GENESIS, Exploring Realistic Neural Models with the General Neural Simulation System, 2nd Edition, Springer, New York, 1998. [2] B.P. Burton, T.S. Chow, A.T. Duchowski, W. Koh, B.H. McCormick, Exploring the brain forest, Neurocomputing 26}27 (1999) 971}980. [3] D.J. Fellman, D.C. Van Essen, Distributed hierarchical processing in the primate cerebral cortex, Cerebral Cortex 1 (1991) 1}47. [4] J.L. Gross, J. Chen, Algebraic speci"cation of interconnection network relationship by permutation voltage graph mappings, Math. Systems Theory 29 (1996) 451}470. [5] C. Harris, Presentation at the Society for Neuroscience Conference, 1998. [6] C.C. Hilgetag, G.P.C. Burns, M.A. O'Neill, M.P. Young, Cluster structure of cortical systems in mammalian brains, in: J. Bower (Ed.), Computational Neuroscience: Trends in Research, 1998, Plenum Press, New York, 1998. [7] W. Koh, B.H. McCormick, A web-based distributed database for brain microstructure, Neurocomputing, 2000, this issue.
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[8] B.H. McCormick, Design of a brain tissue scanner, Neurocomputing 26}27 (1999) 971}980. [9] B.H. McCormick, G.T. Prusky, S. Tewari, Stochastic modeling of the pyramidal cell modules, in: J.M. Bower (Ed.), Computational Neuroscience: Trends in Research 1997, Plenum Press, New York, 1997, pp. 129}134. [10] B.H. McCormick, R.W. DeVaul, W.R. Shankle, J.H. Fallon, Modeling neuron spatial distribution and morphology in the developing human cerebral cortex, Neurocomputing (2000), this issue. [11] J.W. Scannell, Determining cortical landscapes, Nature 386 (1997) 452. [12] E.L. White, Cortical Circuits: Synaptic Organization of the Cerebral Cortex Structure, Function, and Theory, Birkhauser, Boston, 1989. [13] K.G. Wilson, Phys. Rev B 4 (1971) 3174, 3184. [14] M.P. Young, J.W. Scannell, M.A. O'Neill, C.C. Hilgetag, G. Burns, C. Blackmore, Non-metric multidimensional scaling in the analysis of neuroanatomical connection data and the organization of the primate cortical visual system, Phil. Trans. R. Soc. London 3 (348) (1995) 281}308. [15] M.P. Young, J.W. Scannell et al., Analysis of connectivity: neural systems in the cerebral cortex, Rev. Neurosci. 5 (1994) 227}250.
Bruce H. McCormick is Professor of Computer Science and the Director of the Scienti"c Visualization Laboratory at Texas A&M University. His research interests include scienti"c visualization, brain mapping, computational neuroscience, and neural networks. He received his B.S. and Ph.D. degrees in Physics from MIT and Harvard University, respectively. He was Professor of Computer Science and Physics at the University of Illinois at Urbana-Champaign. At the University of Illinois at Chicago, he served as head of the Department of Information Engineering, and at Texas A&M University, as the "rst head of the Department of Computer Science.
Wonryull Koh received a B.S. in Computer Science and Mathematics from the University of Texas at Austin. She has a M.S. in Computer Science from Texas A&M University. Her research interests include computer graphics and scienti"c visualization. Her thesis research formulates a structural information framework for the human neocortex.
William R. Shankle is an Associate Professor of Cognitive Science at UC Irvine. He was trained as a statistician and neurologist. His research focuses on data analysis of brain development, aging and degeneration. With BH Landing, he overturned the dogma of no postnatal neurogenesis in human cerebral cortex.
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James H. Fallon is Professor of Anatomy and Neurobiology in the College of Medicine at the University of California, Irvine. His areas of interest are in the study of the function of neurotrophic factors in neurodegenerative disorders, aging and development, and their use in neural stem cell therapies. He was the "rst to localize a characterized growth factor in the brain, and the "rst to localize EGF, TGF, and a FGF in the brain. He has a long-standing interest in neurochemical anatomy of monoamine, opioid, and other neurotransmitter systems in the mammalian brain. He has collaborated extensively as the systems neuroanatomist on numerous PET and MRI imaging studies in human. He is now collaborating with Junko Hara and Rod Shankle on analyses of developing human cortex, as well as novel analyses of art and the artist's brain, and with these scientists and Bruce McCormick on geometric modeling of human cortex.
Geometric modeling of local cortical networks B.H. McCormick *, W. Koh , W.R. Shankle, J.H. Fallon Scientixc Visualization Laboratory, Department of Computer Science, Texas A&M University, College Station, TX 77843-3112, USA Department of Cognitive Sciences, University of California, Irvine, CA 92697, USA Department of Anatomy and Neurobiology, University of California, Irvine, CA 92697, USA Accepted 13 January 2000
Abstract Geometric models of local cortical microstructure can be used to develop neuroanatomically based network models. First, neuron/"ber spatial distributions and neuron morphology, drawing on a new microstructure imaging technology, must be quanti"ed within the cortical tissue. Given this database, stochastic models recreating the spatial distribution and morphology of the observed neurons are constructed. Synthetic neuron forests are then stochastically generated. Neurite near-collisions (e.g., between axons and dendrites) are modeled to add synaptic connections. The synapse estimation complements the forest of synthetic neurons by adding local connectivity, i.e., wiring up the forest into network models. 2000 Published by Elsevier Science B.V. All rights reserved. Keywords: Cortex; Neuroanatomy; Brain connectivity; Hierarchical cortical network models; Stochastic modeling of synapse formation
1. Introduction The connectivity of human and other mammalian brains is largely uncharted. In our time, large-scale telescopes have opened the external universe to measurement and modeling of its evolution. In the same way, new large-scale light microscopes, processing teravoxels of image data per day [8], promise to open the internal connectivity of brains of all species, including human, to measurement and modeling
* Corresponding author. Tel.: #1-979-845-8870; fax: #1-979-847-8578. E-mail address: [email protected] (B.H. McCormick). 0925-2312/00/$ - see front matter 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 2 3 1 2 ( 0 0 ) 0 0 2 0 0 - 9
462
B.H. McCormick et al. / Neurocomputing 32}33 (2000) 461}469
of brain development and evolution, a natural precursor to detailed functional modeling of the brain. Anticipating this technological development, we describe below a new methodology that recasts three-dimensional reconstruction of brain microstructure into models of local cortical networks. This will allow computational neuroscientists to incorporate neuroanatomical realism into their models. 1.1. Global connectivity databases and their analysis Enno-Stefan and others at the C&O Vogt Institute in Germany and at Newcastleupon-Tyne University in England have collated hundreds of studies on the connectivity of the macaque monkey cerebral cortex [15,14,11,6]. They have created a database that allows one to de"ne the known connectivity, their level of certainty and resolution, and relate them across di!erent cytoarchitectural classi"cation schemes. This invaluable resource represents a set of hypotheses for the connectivity of other primates, including Homo Sapiens. As such, they will be used to compare interspecies cortical connectivity maps as well as to "ll in missing information until human data become available. 1.2. Modeling local connections vs. long-range connections Modeling long-range connections (i.e., association "bers) will be much more di$cult than modeling short-range connections. As a general principal, the serial progression of cortico-cortical projections from primary to secondary to tertiary to higher-order association cortices is known. Nonetheless, at a "ner level of detail, the patterns of association "bers are highly speci"c, arranged in a myriad of parallel projections, and do not follow any obvious rules. The factors that lead some pairs of areas to form connections over considerable distances (e.g., from occipital to frontal cortex), whereas others do not, are not understood. Until they have been identi"ed, modeling (other than database modeling) of long-range connections may prove intractable. For this reason the geometric models of cortical microstructure proposed here focus on local connectivity.
2. Hierarchical network models of brain connectivity 2.1. What constitutes brain connectivity? The brain can be represented as a collection of interconnected neurons. Brain connectivity can be described at four levels of detail (LOD). Our approach is to carefully characterize brain connectivity at the level of volume-"lling circuitry, and describe pooling strategies to derive characterizations of brain connectivity at coarser levels of detail. The contribution of glial cells, blood vasculature, the ventricles, and the like will be ignored below.
B.H. McCormick et al. / Neurocomputing 32}33 (2000) 461}469
463
2.2. Levels of detail E Volume-xlling circuitry (exhibiting cell-to-cell connectivity) records which neuron (and type) is the source of the axon, where the axonal arbor projects (into which pools of neurons); and for each of its axonal segments, ascertains the type and location of all its synapses on postsynaptic neurons. E Population-based connectivity (from neuron morphology and "ber tract modeling). Population-based connectivity modeling aims to capture the local circuitry in a manner that may be generalized over a small population of source and target neurons and their mutual connections. E Finite element modeling of brain connectivity, where discrete bundles of "bers reach across the brain to link the hexahedral elements in a "nite element model of the brain. A typical Brodmann area in human cerebral cortex is modeled by a threedimensional mesh of 50 elements (4 mm;4 mm;3 mm), and similar numbers are needed for other brain nuclei. Finer grain may be required for the resolution of subnuclei. E Global connectivity at a nucleus/cortical area level (e.g., Fellman and Van Essen's hierarchical connection matrix for monkey visual cortex [3]; the C&O Vogt ORT Connectivity Database; G. Burns's connectivity database for the rat brain). 2.3. Brain tissue connectivity at a volume-xlling level of detail Connections are mediated through synapses [12]. A synapse is a connection between a presynaptic and a postsynaptic process (e.g., between a source axon and a target dendrite). Envision the following gedanken experiment: each synapse is cut and split into its pre- and post-synaptic terminals. These terminals are labeled with a universal synaptic address and then assigned, respectively, to their pre- and postsynaptic neurons. This information completely describes the neural connectivity of the tissue: its neural network can be reconstructed solely from a description of its individual neurons.
3. Web-based description of brain tissue The Web-based implementation of our distributed database for brain microstructure [7] includes an exploratory environment, an Intranet, in which to model the pooling strategies needed to characterize local cortical connectivity at coarser LOD than the ideal level, namely, volume-"lling circuitry. The interpretation is both natural and straightforward. Web pages are assigned to all "les, and in particular, to the synapse "les described in the distributed database system. The user, examining a given dendritic segment of the neuron, can open the synapse page for the segment and select a postsynaptic terminal. A mouse click links the user to the associated axonal segment page of the presynaptic neuron, and identi"es the associated presynaptic terminal. Similar strategies can be applied to axonal segments and soma.
464
B.H. McCormick et al. / Neurocomputing 32}33 (2000) 461}469
4. Construction of network models based on the neuroanatomy 4.1. Estimating synapses from neurite near-collisions In principle, the synaptology at a volume-xlling circuitry level of detail of all neurons within a "nite element can be estimated. How is this possible? Consider the following gedanken experiment: generate a synthetic forest of neurons "lling the speci"ed "nite element, where the forest preserves the spatial distributions (layer-by-layer) and neuron morphologies observed within the "nite element [10]. Then examining each synthetic neuron in turn, evaluate the probability that its soma (considered as a segment at level 0) and higher-level dendritic segments form potential synapses with neighboring "bers. These neighboring "bers, candidates for synapse formation, can be axonal segments of neighboring neurons, recurrent axon collaterals, or association "bers. For each "ber there is a point of closest approach by a dendritic segment, and it is at this point that a synapse is most likely to form (Fig. 1). To complete the process we need to invoke the `principle of the promiscuous neurona, to wit `if a dendrite and a "ber can form a more perfect union (a synapse), they willa. Clearly, the probability of a synaptic union drops o! sharply with the inter-segment distance r. But within 1}4 lm, the probability of spines developing and synapses forming is high [12]. Christine Harris, in her recent lapsed-time video of synapse formation in cultured neurons [5], shows budding spines `gropinga for one another. So, quite as growthbased modeling of neuron morphology is driven by knowledge of growth cone dynamics, the synaptology of brain tissue will be driven by knowledge of spine dynamics and the groping process by which synapses are formed. What we have shown above is that, in principle, we could "ll out in complete detail, including its synaptology, the distributed database of neuron morphology.
Fig. 1. Point of closet approach of dendritic segment to an axial "ber (axon) [9].
B.H. McCormick et al. / Neurocomputing 32}33 (2000) 461}469
465
Furthermore, the resulting distribution of synapses could be con"rmed by synapsebased staining. 4.2. Geometric models of local cortical networks What is required to make the above process computationally tractable? First, the synthetic neurons of the "nite element are partitioned into classes (population pools) of known probability. Neurons are spatially partitioned by the position of their soma into concentric stacks of wafers, each wafer modeling one cortical layer centered around a given vertical axis (Figs. 2 and 3). Then, the synthetic neurons within each wafer are partitioned by cell morphology into disjoint "ner-grained subclasses, from which prototypical neurons are selected. Next, the spatial distribution of "bers (both recurrent axon collaterals and association "bers) that can impinge upon the dendritic arbors of these prototypical neurons is similarly modeled. By this process the immense number of neurons (and axonal "bers) in the "nite element has now been reduced to a tractable number of prototypical neurons and "bers. Using these prototypes, the appropriate spatial distribution of dendritic segments and "bers is estimated by a standard Monte Carlo simulation. Though not universally applicable, translational and rotational invariance can be evoked to simplify the spatial distribution functions [9]. This procedure sketches the methodology to estimate the intrinsic connectivity in the developing human cerebral cortex. Models of the human cerebral cortex will be needed for all cortical areas, agepoints, and neurological diagnoses. These models, in fact, describe not only the
Fig. 2. Modeling environment for cylindrical module of cells [9].
466
B.H. McCormick et al. / Neurocomputing 32}33 (2000) 461}469
Fig. 3. Interaction at the wafer level [9].
population-based local connectivity of the tissue, but also describe canonical cortical models of the neural circuitry. To our knowledge, these methods are new and described here for the "rst time. Constructing anatomical-based neural microcircuits in this way}driven directly from measurements of the cortical microstructure represents a signi"cant advance in computational neuroscience.
4.3. Finite element modeling of brain connectivity Consider all neurons of a given type whose soma resides in a "nite element at a given locus in the cerebral cortex [2]. Imagine that by injecting a bolus of stain into the "nite element that the soma, dendritic arbors, and axonal arbors of all these cells, and only these cells, were completely "lled with stain. Changing from a neuronal perspective to a "nite element level of detail, we now pool the neurons of the "nite element together to derive a pooled description of brain connectivity. Speci"cally, we visualize the "nite element as the `somaa of a `fat neurona, which in general has discrete bundles of association "bers reaching out across the brain. What, at a neuronal level of detail, were synapses between an individual association "ber and the dendrites and soma of its target neurons become, at this coarser level of description, `fat synapsesa between the bundles of association "bers and the "nite elements housing the target neurons. This example shows that to describe brain connectivity at a "nite element level of detail (or, more generally, at any of the three lower levels of detail) we must establish a formal morphological description of these "nite element (population-based) pools. And replacing the synapse, we must establish conventions for what constitutes a `synapsea at this coarser level of description. In e!ect, the pooled neurons, considered as the sum of their parts, carry too much detailed information. Our task is to "nd what parts of this information can be thrown away without jeopardizing the description of brain connectivity at lower levels of detail.
B.H. McCormick et al. / Neurocomputing 32}33 (2000) 461}469
467
4.4. Hierarchical network structure and gauge theory Gross and Chen [4] have developed a systematic approach to manipulate hierarchical network structures. In this approach, large-scale networks with certain symmetry are represented in a very compact manner that greatly reduces the computational complexity. This approach "ts very well our characterization of brain connectivity described at the four levels of details, which form a hierarchy of networks. Moreover, we are investigating the relation between this representation and gauge theory in modern particle physics and condensed matter physics [13]. Gauge theory is a powerful mathematical tool in the study of such hierarchical networks.
5. Interface to multi-compartmental modeling of cortical microcircuits The overriding objective in these studies is to create canonical geometric models of local cortical microstructure, by cytoarchitectural area, age and neurological diagnosis, that can be used to develop neuroanatomical-based cortical circuit models. We propose to provide an interface to GENESIS [1], a widely used neural simulator, to facilitate this transition from geometric models of local cortical microstructure to dynamic modeling of the neuroanatomical-based cortical circuits.
Acknowledgements This work was supported in part by Texas Advanced Technology Program Grant 999903-124 (McCormick) from the Texas Higher Education Coordinating Board.
References [1] J. Bower, D. Beeman, The Book of GENESIS, Exploring Realistic Neural Models with the General Neural Simulation System, 2nd Edition, Springer, New York, 1998. [2] B.P. Burton, T.S. Chow, A.T. Duchowski, W. Koh, B.H. McCormick, Exploring the brain forest, Neurocomputing 26}27 (1999) 971}980. [3] D.J. Fellman, D.C. Van Essen, Distributed hierarchical processing in the primate cerebral cortex, Cerebral Cortex 1 (1991) 1}47. [4] J.L. Gross, J. Chen, Algebraic speci"cation of interconnection network relationship by permutation voltage graph mappings, Math. Systems Theory 29 (1996) 451}470. [5] C. Harris, Presentation at the Society for Neuroscience Conference, 1998. [6] C.C. Hilgetag, G.P.C. Burns, M.A. O'Neill, M.P. Young, Cluster structure of cortical systems in mammalian brains, in: J. Bower (Ed.), Computational Neuroscience: Trends in Research, 1998, Plenum Press, New York, 1998. [7] W. Koh, B.H. McCormick, A web-based distributed database for brain microstructure, Neurocomputing, 2000, this issue.
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[8] B.H. McCormick, Design of a brain tissue scanner, Neurocomputing 26}27 (1999) 971}980. [9] B.H. McCormick, G.T. Prusky, S. Tewari, Stochastic modeling of the pyramidal cell modules, in: J.M. Bower (Ed.), Computational Neuroscience: Trends in Research 1997, Plenum Press, New York, 1997, pp. 129}134. [10] B.H. McCormick, R.W. DeVaul, W.R. Shankle, J.H. Fallon, Modeling neuron spatial distribution and morphology in the developing human cerebral cortex, Neurocomputing (2000), this issue. [11] J.W. Scannell, Determining cortical landscapes, Nature 386 (1997) 452. [12] E.L. White, Cortical Circuits: Synaptic Organization of the Cerebral Cortex Structure, Function, and Theory, Birkhauser, Boston, 1989. [13] K.G. Wilson, Phys. Rev B 4 (1971) 3174, 3184. [14] M.P. Young, J.W. Scannell, M.A. O'Neill, C.C. Hilgetag, G. Burns, C. Blackmore, Non-metric multidimensional scaling in the analysis of neuroanatomical connection data and the organization of the primate cortical visual system, Phil. Trans. R. Soc. London 3 (348) (1995) 281}308. [15] M.P. Young, J.W. Scannell et al., Analysis of connectivity: neural systems in the cerebral cortex, Rev. Neurosci. 5 (1994) 227}250.
Bruce H. McCormick is Professor of Computer Science and the Director of the Scienti"c Visualization Laboratory at Texas A&M University. His research interests include scienti"c visualization, brain mapping, computational neuroscience, and neural networks. He received his B.S. and Ph.D. degrees in Physics from MIT and Harvard University, respectively. He was Professor of Computer Science and Physics at the University of Illinois at Urbana-Champaign. At the University of Illinois at Chicago, he served as head of the Department of Information Engineering, and at Texas A&M University, as the "rst head of the Department of Computer Science.
Wonryull Koh received a B.S. in Computer Science and Mathematics from the University of Texas at Austin. She has a M.S. in Computer Science from Texas A&M University. Her research interests include computer graphics and scienti"c visualization. Her thesis research formulates a structural information framework for the human neocortex.
William R. Shankle is an Associate Professor of Cognitive Science at UC Irvine. He was trained as a statistician and neurologist. His research focuses on data analysis of brain development, aging and degeneration. With BH Landing, he overturned the dogma of no postnatal neurogenesis in human cerebral cortex.
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James H. Fallon is Professor of Anatomy and Neurobiology in the College of Medicine at the University of California, Irvine. His areas of interest are in the study of the function of neurotrophic factors in neurodegenerative disorders, aging and development, and their use in neural stem cell therapies. He was the "rst to localize a characterized growth factor in the brain, and the "rst to localize EGF, TGF, and a FGF in the brain. He has a long-standing interest in neurochemical anatomy of monoamine, opioid, and other neurotransmitter systems in the mammalian brain. He has collaborated extensively as the systems neuroanatomist on numerous PET and MRI imaging studies in human. He is now collaborating with Junko Hara and Rod Shankle on analyses of developing human cortex, as well as novel analyses of art and the artist's brain, and with these scientists and Bruce McCormick on geometric modeling of human cortex.