Giorgio A. Ascoli*, Lawrence Hunter, Jeffrey L. Krichmar, James L. Olds and Stephen L. Senft

Krasnow Institute for Advanced Study at George Mason University, MS2A1 Fairfax VA 22030-4444.
*Email: ascoli@gmu.edu – Ph. (703)993-4383, Fax (703)993-4325

Scientific reports of the peculiar shape of neuronal arborizations date back to the work of such pioneers as Purkinje and Golgi. Almost a century ago, Ramon y Cajal proposed that the great variety of dendritic morphology shown by different neuronal families possibly reflected and affected different physiological functions. Particularly, the mammalian hippocampus (Fascia Dentata and Cornus Ammoni) has been traditionally the focus of attention for many neuroscientists aiming at understanding the role of anatomy in nervous system functions. The reason for this peculiar interest in the hippocampus is two-fold. First, the role of the hippocampal formation (hippocampus proper, entorhinal cortex, and septal fibers) in the neuronal mechanisms of associative learning is relatively well established, at least compared to the obscure functions of most other vertebrate brain regions. Second, the anatomy of the hippocampus proper is relatively ordered and simple, and soon became paradigmatic of structural principle of nervous organization, from histology to morphology. This wealth of data, from behavior to electrophysiology and quantitative imaging at the cellular level, is an outstanding opportunity for computational neuroscience. The development of virtual reality techniques, and the breath-taking progress of computer graphics now allow for the implementation of extremely complex algorithms to create, display, and manipulate realistic networks of detailed neurons. Here, we apply some of the tools we have been developing to a neuroanatomical model of the hippocampus. Fifteen classes of neurons, distinct for morphology and or location, are modeled in detail, and synaptic connections are created in a specific fashion (axo-somatic, axo-dendritic and axo-axonic). Excitatory, inhibitory, and modulatory connections are modeled, and both principal cells and interneurons are included. The key characteristic of this approach is its stochastic / statistical definition. All parameters are used as distributions of values, which means that no two networks, or single neurons in a network, created with the same parameters are identical (unless the random generator seed is locked). Positive long-term side effect results of this research include a great potential for data compression and a powerful mean of data visualization both for scientific and educational purposes.
 
 

Figure 1: The hippocampal formation. (Upper panel adapted from Schultz, Panzeri, Rolls, and Treves: Quantitative analysis of a Schaffer collateral model. In Information Theory and the Brain, Baddeley, Hancock, Foldiak eds., Cambridge University Press, UK, 1998). The entorhinal cortex (EC), modeled as black box columns, sends perforant pathway fibers (PP) to the dentate gyrus (DG) and to CA3. DG granule cells output mossy fibers (MF) to CA3. CA3 pyramidal cells send axons recurrently into CA3 and to CA1 through the Schaffer collaterals (SC). CA1 pyramidal cells project back to EC (and to the subicular complex, not modeled). Principal cells of DG, CA3 and CA1 also receive cholinergic input from the medial septal complex (not modeled) via the septo-hippocampal pathway (SHP), modeled as a synchronous input. All the cells and the connections within DG, CA3 and CA1 are modeled in detail (lower panel): the flat scheme of the hippocampus shows the typical three-layer archicortical organization. The DG is divided into hilus/polymorphic layer (H), granule cell layer (G), and molecular layer/fascia dentata (M), which contains the granule cell dendrites. The CA fields are divided into lacunosum layer/stratum radiatum (L), which contains the pyramidal cell basal dendrites, pyramidal cell layer, and alveus/stratum radiatum (A), which contains the pyramidal cell apical dendrites. Fifteen distinct classes of neurons are to be modeled, specified in each layer of the bottom panel: gabaergic polymorphic (gpc), mossy (mc), chandelier (cc), granule, basket, and molecular perforant pathway associated (mopp) cells in the dentate gyrus; radiatum (ri) and oriens (oi) interneurons, and chandelier and pyramidal (pc) cells in CA3; lacunosum-moleculare (lm) and oriens/alveus (oa) interneurons, and chandelier, pyramidal and basket cells in CA1. For a hyperlinked version of this scheme, see www.krasnow.ascoli/hipproject/main.html.

Table 1: Numbers of neurons in the hippocampus and in the model. The choice of the neuronal classes to be modeled was based on known anatomical and physiological characteristics and on the availability of quantitative data. The numbers of neurons and their connectivity matrix has been estimated in four recent fundamental reviews of the literature (N/A: not applicable). All the experimental data focus on the rat hippocampus, and that is therefore the structure we chose to model. Amaral, Ishizuka and Claiborne critically reviewed the available evidence on hippocampal cell counting (Neurons, numbers and the hippocampal network. Progr. Brain Res. 83:1-11, 1990). Freund and Buzsaki massively examined all quantitative aspects of inhibitory connections (Interneurons of the hippocampus. Hippocampus 6:347-470, 1996). Bernard and Wheal summarized the synaptic pathways between CA3 and CA1 (Model of local connectivity patterns in CA3 and CA1 areas of the hippocampus. Hippocampus 4:497-529, 1994), while Patton and McNaughton completed the picture with the dentate gyrus (Connection matrix of the hippocampal formation: I. The dentate gyrus. Hippocampus 5:245-286, 1995). Aside from the cortical columns of the entorhinal regions (that are modeled as black boxes, like the septohippocampal connection), a realistic model of the hippocampis would contain approximately 1,600,000 neurons. In order to reduce this number to a more tractable system, we consider a transversal section of the hippocampus and adopt a typical exponential scheme of scaling. In this case, we imposed Nm=a*Nr^b, where Nm and Nr are the number of cells in a specific class in the real hippocampus, Nm is the corresponding number in the simulated model, and a and b are numeric constants set respectively at 0.004 and 0.8. Number of synapses were calculated approximately with the radical law Sm/Sr=sqr(Nm/Nr), (where Sm and Sr are the modeled and real number of synapses, respectively), and then doubled, in order to allow for future network model upscaling. The number of compartments per cell and per field (Ccell and Cfield, respectively) was estimated as identical to the number of synaptic contacts (i.e. one input per compartment). Inhibitory interneurons, which constitute less than 4% of the total cells in the real hippocampus, sum up to over 7% in the simulated network model, due to the exponential scaling scheme. However, because of the extensive divergent pattern of connectivity of interneurons (each of them typically inhibits thousand of principal cells in the hippocampus), this slight unbalance is not expected to alter significantly network dynamics.

Figure 2: Quantitative morphological data. In order to model hippocampal neurons in an anatomical accurate and realistic fashion, we need quantitative, three-dimensional data for all the cells involved in the model. We divided the fifteen cell classes into five morphologically distinct families: granule cells, CA3 pyramidal cells, CA1 pyramidal cells, pyramidal-like interneurons (basket and chandelier cells, radiatum and lacunosum-moleculare interneurons), and polymorphic interneurons (mossy cells, gabaergic polymorphic cells, oriens and oriens-alveus interneurons). Some of these families are further subdivided depending on the axonal patterns (e.g. pyramidal-like interneurons are divided into axo-axonal/chandelier-, axo-somatic/basket, or axo-dendritic/radiatum cells). The quantitative experimental data corresponding to these morphological classes can be obtained through electronic archives available through the internet. The main archives are produced by the groups of Turner at the Universities of Duke and Southampton (http://www.neuro.soton.ac.uk/cells/cellArchive.html), Amaral at the University of California ad Davis (ftp://mossycell.ucdavis.edu/public), and Claiborne at the University of Texas at San Antonio (ftp://harlan.ls.utsa.edu/pub/). The Southampton archive is so far the richest and most complete hippocampal resource, as it contains neurons from all morphological classes. It also distributes its own neuronal viewer, both as Java applet and as Dos utility. Turner’s data are published with Pyapali and coworkers (e.g. Dendritic properties of hippocampal CA1 pyramidal neurons in the rat: intracellular staining in vivo and in vitro. J. Comp. Neurol. 391:335-352, 1998, and therein references to previous work). Claiborne’s own morphometric analysis on dentate gyrus granule cells is also published with coworkers (Quantitative, three-dimensional analysis of granule cell dendrites in the rat dentate gyrus. J. Comp. Neurol. 302:206-219, 1990). A hyperlinked version of each cell class’ connectivity scheme is available at http://www.krasnow.gmu.edu/ascoli/hipproject/main.html.

Figure 3: Amaral’s anatomical data. Amaral’s electronic morphological archive, available at Davis (data published and critically analyzed by Ishizuka et al. in A quantitative analysis of the dendritic organization of pyramidal cells in the rat hippocampus. J. Comp. Neurol. 362:17-45, 1995) is particularly valuable because several neurons (i.e. CA3 and CA1 pyramidal cells) are stored in datafiles also containing fiduciary axes, such as those shown in this figure. The exact position of the cells in the layers can be then established precisely, and possible subtle morphological differences in the subfields noticed and included in the model (in the picture, a CA1 pyramidal cell at the boarder with CA3). An additional advantage of this archive is that axonal initial segments are usually included in the morphological structures. Amaral’s datafiles are in standard Cartesian coordinates (compartmentID, X, Y, Z, diameter, parentID), and can be read in and displayed by the software package ArborVitae developed by Senft (A statistical framework to present developmental neuroanatomy. In Biobehavioral Foundations, Donahoe, J., Ed., Elsevier Press, 1997). Other useful quantitative morphological data on hippocampal neurons are published by Desmond and Levy (A quantitative anatomical study of the granule cell dendritic fields of the rat dentate gyrus using a novel probabilistic method. J. Comp. Neurol. 212(2):131-145, 1982, and unpublished results, personal communication). Altogether, neurons in electronic www databases amount to more than 200 cells for the hippocampal formation of the rat. However, archives are growing quickly and we expect this number to double within the next few years.

Figure 4: From real neurons to virtual analogs. One of the most peculiar characteristics of our hippocampal model is to consider the morphological details of the single-cell components. About 500 neurons belonging to five morphologically distinct categories and divided into 15 anatomical classes are modeled with precise Cartesian compartments and connectivity. The approach for this massive model is stochastic and statistical, as described previously (Senft, 1997; Ascoli et al., Computer generation of sets of anatomically plausible neurons for modeling. J. Neurosci. (Suppl.) 645.5, 23:1674, 1997). The general approach can be summarized as follows. A set of real neurons available from electronic internet archives (in the figure, the top four CA3 pyramidal cells) are analyzed for fundamental morphological parameters such as average branch length, diameter, taper, appending/extending/branching angles (elevation and azimuth), and so forth. These parameters are then fed into morphogenic algorithms (such as ArborVitae or L-Neuron) under the form of statistical distributions (e.g. average, standard deviation and tails for gaussian distributions, or median and min/max range for uniform distributions). A set of virtual neurons is then created in a stochastic way (in the figure, the bottom four CA3 pyramidal cells, created with ArborVitae). Real and virtual neurons are then compared in terms of standard emergent morphometric parameters such as tree asymmetry, Sholl’s branch distribution patterns etc. Fundamental parameters in virtual neurons are then changed with additional global constraints (e.g. attractor/repulsors, tropism, internal competition) to minimize the discrepancies from the real cells. Color code of the virtual neurons in the figure (input fibers not displayed): basal dendrites (receiving inputs from septal cholinergic afferents, Schaffer glutamatergic collaterals and oriens gabaergic interneurons) are brown, somata (receiving inputs from gabaergic basket cells) are pink, proximal apical dendrites (receiving inputs from glutamatergic dentate mossy fibers Schaffer collaterals) are green, and distal apical dendrites (receiving inputs from glutamatergic entorhinal perforant pathway and Schaffer collaterals, and gabaergic radiatum interneurons) are blue. Axons (receiving gabaergic inputs from chandelier cells) are not shown.
 
 

#	T	SS	BA#	BL	BE#	BT#	BH	BW	BD	BVPA
1	a	1.28	5	7533	1834	52	189	255	123	0.490956
2	a	1.45	1	4017	1167	27	220	192	121	0.674356
3	a	0.51	3	9266	2213	32	197	219	150	0.613346
4	a	0.9	3	5074	1801	31	131	185	168	0.524037
5	a	0.82	3	4706	1783	23	137	227	194	0.469013
6	a	2.15	5	5384	1815	20	191	195	154	0.604545
7	a	0.82	4	4325	1131	24	199	225	118	0.531713
8	a	2.14	7	7194	1440	29	251	167	143	0.295455
9	a	0.27	6	4125	659	25	205	180	54	0.645192
10	a	0.55	4	5661	977	30	218	253	77	0.435789
11	b	0.41	4	6905	1299	34	193	224	161	0.517421
12	b	0.56	7	9501	2012	54	192	283	211	0.600567
13	b	0.26	3	6799	1795	40	218	266	167	0.502223
14	b	0.26	5	6599	1729	44	214	291	199	0.561909
15	b	0.53	3	5827	1530	32	206	275	147	0.365729
16	b	0.54	5	8020	2231	45	211	254	141	0.659702
17	b	0.26	2	3897	625	11	246	287	581	0.592592

Table 2: An example of morphometric analysis. This table shows a typical analysis of algorithmic and morphometric parameters. In this examples, a set of seventeen real CA1 pyramidal cells was considered. For the sake of simplicity, only a few morphological parameters from basal dendrites are reported (analogous ones can be measured for apical dendrites, axons etc.). Neurons were taken from two different electronic databases i.e. Turner’s Duke archive and Amaral’s Davis archive (a and b, respectively, in the column T as in Type). The first four columns report algorithmic parameters (i.e. values that can be used by programs such as ArborVitae to generate virtual neurons): somatic surface area (SS, in 10^-3mm²), number of basal stems (BA#), total dendritic length (BL, in mm), and total number of dendritic segments (BE#). The last five parameters are exemplars of "emergent" morphometric variables: number of termination points (BT#), tree height (BH), width (BW), and depth (BD), all in mm, and van Pelt’s asymmetry (BVPA). Morphometric parameters are not used explicitly or directly in the algorithm, and they can therefore be used to test whether real and virtual neurons are actually similar. For instance, tree asymmetry or partition was introduced by van Pelt’s group to measure the balance of branches (Natural variability in the number of dendritic segments: model-based inferences about branching during neurite outgrowth. J. Comp. Neurol. 387:325-340, 1997). van Pelt’s asymmetry is defined between 0 (totally symmetric tree) and 1 (totally asymmetric tree), and changes sensitively with the topological mechanism of growth. Other useful emergent morphometric parameters have been adopted by Larkman based on Sholl’s radial distributions (Dendritic morphology of pyramidal neurons of the visual cortex of the rat: I. Branching patterns. J. Comp. Neurol. 306:307-319, 1991). In Larkman’s approach, parameters such as the number of dendritic branches are plotted versus a distance from the soma (Euclidean or along the dendritic path). The distributions can be then characterized statistically (average and standard deviation, or median and range, etc.), and the fundamental statistical parameter can be used as quantitative "emergent" morphometric markers of dendritic shape. Dynamical rendering virtual and real neurons in a 3D computer graphic display enormously facilitates the measurements of morphometric parameters, making it possible to analyze completely ten to twenty neurons every day.
 

Figure 5: Creating the network backbone. In the present and next figures, a reduced-scale example of a hippocampal model network is presented. The program ArborVitae was run on a Silicon Graphic O2 workstation with 256 MB of RAM. In this simulation, about 50 neurons and 100,000 compartments are instrumented (the whole implementation takes approximately 100 seconds of clock time on a dedicated machine). The upper panel shows the distribution of neuronal somata. The typically double-C shaped structure of the hippocampus is obtain by encoding separately for the three main subfields, namely dentate gyrus (somata granule cell layer), CA3, and CA1 (somata in pyramidal cell layer). Each set of cell bodies follows a uniform distribution in all three Cartesian coordinates. The three sets are then independently curved in a sinusoidal fashion and tilted circularly along one of the axes to obtain a good fit with experimental data (from the rat brain atlas). The resulting structure has each individual cell oriented in the appropriate direction, and distributed laminarly in a very thin transversal section (2-3 somata of thickness). At this stage, dendrites are stemmed from the somata (lower panel). Neurons are not appropriately in scale in this figure, and CA3 and CA1 pyramidal cells have a very narrow and similar distribution of morphometric parameters. Nonetheless, the typical anatomy of the main stages of hippocampal formation can be well recognized (granule cells are purple, CA3 pyramidal cells are brown, CA1 pyramidal cells are green). No two neurons are exactly alike, given the stochastic and statistical nature of the generative algorithm. Similarly, the program will produce a different network (though with the same overall anatomical characteristics) at each re-run.
 

Figure 6: The complete small-scale network. In this screenshot, an image of the complete network is presented. While interneurons are still missing from the picture, axons and synaptic connections have been added to the previous backbone. On the extreme left and right of the network, the septohippocampal pathway and the entorhinal cortical columns are modeled as black boxes (with one and 10 compartments, respectively, in this model). The septohippocampal pathway sends its fibers onto the basal dendrites of CA3, while entorhinal perforant pathway axons make synapses into the molecular layer of the dentate gyrus (on dendritic compartments of granule cells). Both these "external" projections are colored in light blue. "Internal" hippocampal projections include mossy fiber axons (indigo color) from dentate granule cells to apical dendrites of CA3 pyramidal cells; Schaffer collaterals (purple), which contact apical dendrites of both CA3 and CA1 pyramidal cells; and CA1 efferent axons (yellow), which synapse onto basal CA1 pyramidal cells on their way to contacting entorhinal cortical columns. Backpropagating fibers, inter-commisural pathways, and subicular connections are not modeled in this network. Axonal growth is guided by means a matrix of "attractors", which act as global constraints on top of the morphological distributions of statistical values. Axons are driven through dendritic fields by a blending of their intrinsic growth mode and attraction (or repulsion) by a stochastically sampled sub-set of target compartments. Group competition and spatial inhibition are also implemented on a population basis. When an axon "growth cone" senses multiple attractors from different directions (likely signifying it entered the appropriate dendritic field, rather than it is still traveling long-distance towards the target field), a mechanism of "sprouting, or multiple branching, is implemented, to increase the potential candidates to a presynaptic connections. A synapse is formed when an axonal varicosity and a dendritic spine (or shaft area) fall within a threshold distance (this algorithmic parameter can vary depending on the axonal group identity). After the network is grown and connected, all compartments that do not receive or establish a synaptic contact can be "pruned back" in order to gain in computational efficiency.
 

Figure 7: Reverberating activity in the hippocampal network. Passive current propagation (compartmental "leak") can be simulated online in our modeling environment. In this case, the synchronous depolarization of the cholinergic septohippocampal pathway axons caused the activation of CA3 pyramidal cells, and eventually led the whole network in a stable attractor state. Depending on the number of synapses formed between hippocampal subfields, a specific pattern of injected activity can fade out, saturate the network, or cause a stable reverberation. Although this kind of electrophysiological simulation is not biologically meaningful (as it does not model any active conductance), it shows the potential of advanced graphic display and a joint development of physiological and morphological approach in computational neuroscience. Anatomical files corresponding to the whole simulated network can be saved in a compartmental fashion, listing the parent-to-child relationships, the synaptic connectivity, and the Cartesian locations and size of each cylindrical compartments. An accurate off-line simulation can be then run with standard tools such as GENESIS, Neuron, Surf-hippo, or QRN. One of the most crucial issues, when we consider simulating a 500,000 compartment network model, is computational efficiency. The result, in terms of membrane potential, ionic concentrations, synaptic transmission, somatic spiking behavior etc., can be used for a massive and quantitative data analysis, or it can be re-loaded into the original software package (ArborVitae) for a direct display. In this example, lighter colors indicate larger membrane depolarization (white compartments are near signal saturation).
 

Figure 8: A "fractal" close-up. Virtual reality rendering allows one to move, rotate, zoom, and color-code parts or the entirety of a network. This allows a close inspection of membrane activity, morphological details, and patterns of synaptic connectivity or activation. In this example, the apical dendrites of a CA3 pyramidal cell (brown), invaded by a varicose axonal population (purple), is displayed. To the best of our knowledge, the work presented in this paper constitutes the first example of a computational neuroanatomical model of the rat hippocampus with such a level of morphological detail. Much attention is being paid to hippocampal anatomy in the experimental literature. This result represents one of the first opportunities to instrument a computational model with both realistic physiology and morphology, thus allowing a large scale, biologically plausible, accurate, and detailed network simulation of hippocampal activity.