Ramon Huerta, Lev Tsimring and Ivan Soltesz Jonas Dyhrfjeld

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Ramon Huerta, Lev Tsimring and Ivan Soltesz Jonas Dyhrfjeld

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Jonas Dyhrfjeld-Johnsen, Vijayalakshmi Santhakumar, Robert J. Morgan, Ramon Huerta, Lev Tsimring and Ivan Soltesz
J Neurophysiol 97:1566-1587, 2007. First published Nov 8, 2006; doi:10.1152/jn.00950.2006
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J Neurophysiol 97: 1566 –1587, 2007. First published November 8, 2006; doi:10.1152/jn.00950.2006.
Topological Determinants of Epileptogenesis in Large-Scale Structural and Functional Models of the Dentate Gyrus Derived From Experimental Data

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Jonas Dyhrfjeld-Johnsen,1,* Vijayalakshmi Santhakumar,1,* Robert J. Morgan,1 Ramon Huerta,2 Lev Tsimring,2 and Ivan Soltesz1 1Department of Anatomy and Neurobiology, University of California, Irvine; and 2Institute for Nonlinear Science, University of California, San Diego, California
Submitted 6 September 2006; accepted in final form 5 November 2006

Dyhrfjeld-Johnsen J, Santhakumar V, Morgan RJ, Huerta R, Tsimring L, Soltesz I. Topological determinants of epileptogenesis in large-scale structural and functional models of the dentate gyrus derived from experimental data. J Neurophysiol 97: 1566 –1587, 2007. First published November 8, 2006; doi:10.1152/jn.00950.2006. In temporal lobe epilepsy, changes in synaptic and intrinsic properties occur on a background of altered network architecture resulting from cell loss and axonal sprouting. Although modeling studies using idealized networks indicated the general importance of network topology in epilepsy, it is unknown whether structural changes that actually take place during epileptogenesis result in hyperexcitability. To answer this question, we built a 1:1 scale structural model of the rat dentate gyrus from published in vivo and in vitro cell type–specific connectivity data. This virtual dentate gyrus in control condition displayed globally and locally well connected (“small world”) architecture. The average number of synapses between any two neurons in this network of over one million cells was less than three, similar to that measured for the orders of magnitude smaller C. elegans nervous system. To study how network architecture changes during epileptogenesis, long-distance projecting hilar cells were gradually removed in the structural model, causing massive reductions in the number of total connections. However, as long as even a few hilar cells survived, global connectivity in the network was effectively maintained and, as a result of the spatially restricted sprouting of granule cell axons, local connectivity increased. Simulations of activity in a functional dentate network model, consisting of over 50,000 multicompartmental singlecell models of major glutamatergic and GABAergic cell types, revealed that the survival of even a small fraction of hilar cells was enough to sustain networkwide hyperexcitability. These data indicate new roles for fractionally surviving long-distance projecting hilar cells observed in specimens from epilepsy patients.
The dentate gyrus, containing some of the most vulnerable cells in the entire mammalian brain, offers a unique opportunity to investigate the importance of structural alterations during epileptogenesis. Many hilar cells are lost in both humans and animal models after repeated seizures, ischemia, and head trauma (Buckmaster and Jongen-Relo 1999; Ratzliff et al. 2002; Sutula et al. 2003), accompanied by mossy fiber (granule cell axon) sprouting. In temporal lobe epilepsy, loss of hilar neurons and mossy fiber sprouting are hallmarks of seizureinduced end-folium sclerosis (Margerison and Corsellis 1966; Mathern et al. 1996), indicating the emergence of a fundamentally transformed microcircuit. Because structural alterations in
* These authors contributed equally to this work. Address for reprint requests and other correspondence: J. DyhrfjeldJohnsen, Department of Anatomy and Neurobiology, University of California, Irvine, CA 92697-1280 (E-mail: [email protected]).

experimental models of epilepsy occur concurrently with multiple modifications of synaptic and intrinsic properties, it is difficult to unambiguously evaluate the functional consequences of purely structural changes using experimental techniques alone.
Computational modeling approaches may help to identify the importance of network architectural alterations. Indeed, prior modeling studies of idealized networks indicated the importance of altered network architecture in epileptogenesis (Buzsa´ki et al. 2004; Netoff et al. 2004; Percha et al. 2005). However, to test the role of structural changes actually taking place during epileptogenesis, the network models must be strongly data driven, i.e., incorporate key structural and functional properties of the biological network (Ascoli and Atkeson 2005; Bernard et al. 1997; Traub et al. 2005a,b). Such models should also be based on as realistic cell numbers as possible, to minimize uncertainties resulting from the scaling-up of experimentally measured synaptic inputs to compensate for fewer cells in reduced networks.
Within the last decade, large amounts of high-quality experimental data have become available on the connectivity of the rat dentate gyrus both in controls and after seizures. From such data, we assembled a cell type–specific connectivity matrix for the dentate gyrus that, combined with in vivo single cell axonal projection data, allowed us to build a 1:1 scale structural model of the dentate gyrus in the computer. We characterized the architectural properties of this virtual dentate gyrus network using graph theoretical tools, following recent topological studies of biochemical and social networks, the electric grid, the Internet (Albert et al. 1999; Baraba´si et al. 2000; Eubank et al. 2004; Jeong et al. 2000; Watts and Strogatz 1998), the Caenorhabditis elegans nervous system (Watts and Strogatz 1998), and model neuronal circuits (Lin and Chen 2005; Masuda and Aihara 2004; Netoff et al. 2004; Roxin et al. 2004). To test the functional relevance of the alterations observed in our structural model, we enlarged, by two orders of magnitude, a recently published 500-cell network model of the dentate gyrus, incorporating multicompartmental models for granule cells, mossy cells, basket cells, and dendritically projecting interneurons reproducing a variety of experimentally determined electrophysiological cell-specific properties (Santhakumar et al. 2005).
Taken together, the results obtained from these data-driven computational modeling approaches reveal the topological
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characteristics of the control dentate gyrus and demonstrate that hyperexcitability can emerge from purely structural changes in neuronal networks after loss of neurons and sprouting of new connections, in the absence of changes in synaptic or intrinsic cellular properties.
A three-step strategy was implemented to investigate the functional role of the structural reorganizations that take place in the rat dentate gyrus during epileptogenesis: 1) construction of the database; 2) construction of the structural models (control and diseased versions); and 3) construction of the functional models (control and diseased versions). These three steps will be described first, followed by details of the implementation and assessment of the structural and functional models. Additional details can be found in appendixes A1–A3.
Construction of the database and the models
CONSTRUCTION OF THE DATABASE. The database for the normal and epileptic biological dentate networks was assembled from published data. This process itself entailed several distinct steps. As an initial step, eight types of dentate cells were identified as anatomically well described: granule cells, mossy cells, basket cells, axo-axonic cells, molecular layer cells with axonal projections to the perforant path (MOPP cells), hilar cells with axonal projections to the perforant path (HIPP cells), hilar cells with axonal projections to the commissural-associational pathway (HICAP cells), and interneuron-specific (IS) cells (Fig. 1A). Next, the numbers of cells for each of these eight neuronal types were estimated from the published data (see cell numbers in the left column of Table 1, with references). For a full description of how the cell numbers were estimated, see APPENDIX A1. As a third step in assembling the database, the connectivity matrix was filled in (Table 1). This matrix contains estimates of how many postsynaptic cells among each of the eight cell types a single presynaptic neuron of a given type innervates (for example, from the third row, second column in Table 1: a single basket cell innervates about 1,250 granule cells; mean and ranges are indicated, with references). For full justification of the estimates in the connectivity matrix, see APPENDIX A2. As a final step, spatial constraints in connectivity were considered. For each cell type, the extent of the axons of single cells along the septotemporal axis of the dentate gyrus was determined from in vivo single-cell fills published in the literature (Fig. 2). For example, in the case of control mossy fibers, the averaged in vivo axonal distribution of 13 granule cells (Buckmaster and Dudek 1999) was fitted with a single Gaussian (Fig. 2). For a full description of the construction of the axonal distributions from the in vivo single-cell filling data and the single or double Gaussian fits, see APPENDIX A3.
TION. Once the database was assembled, a structural model of the dentate gyrus was created in the computer. This was a so-called graph network, consisting of “nodes” (corresponding to neurons) and “links” (corresponding to synaptic connections). Each node carried the identity and connectivity pattern of a particular cell type (in other words, there were “granule cell nodes” and “mossy cell nodes,” etc.). The links were directed (like synapses) but nonweighted (meaning that a link simply represented the existence of a connection from cell A to cell B, irrespective of the number of synapses between cells A and B or the functional strength of that connection; note that the functional model, described later, takes some of these factors into account). The structural model was full scale (1:1, meaning that the number of nodes in the graph equaled the total number of cells in the dentate gyrus) and captured the salient connectivity and axonal distribution of the various cell types. Overall, the resulting structural model of the dentate gyrus

was similar to graph representations of other real-world systems (e.g., Watts and Strogatz 1998).
ROSIS. In terms of the structural reorganization of the neuronal networks during limbic epileptogenesis, the loss of hilar cells and the sprouting of mossy fibers are two key factors underlying the process of “end-folium” (meaning the dentate gyrus) sclerosis (Margerison and Corsellis 1966; Mathern et al. 1996) (in the rest of the paper, we will use the shorthand “sclerosis” for end-folium sclerosis; note that end-folium sclerosis is distinct from the broader term “hippocampal sclerosis”). Herein, we simulated the structural changes in sclerosis by removing hilar cells (mossy cells, HIPP cells, HICAP cells, and IS cells) and adding mossy fiber contacts. The biological process of sclerosis (original meaning: “hardening of the tissue”) encompasses more than the loss of cells and sprouting of axons (importantly, it also entails gliosis). However, from the perspective of neuronal network reorganization in the dentate gyrus, the loss of hilar cells and the sprouting of mossy fibers are clearly the two major factors.
There were three important features that needed to be considered during the implementation of sclerosis in the structural model. First, just as in the biological network, the loss of hilar cells entailed the loss of both the excitatory mossy cells and the inhibitory HIPP, HICAP, and IS interneurons in the hilus (Buckmaster and Jongen-Relo 1999). Second, just as in the biological network, the spatial extent of sprouted mossy fibers from a single granule cell remained restricted to a single hippocampal lamella (about 600 ␮m) like the control mossy fibers (Buckmaster et al. 2002b). Third, the progression of sclerosis was implemented by considering full (100%) sclerosis the state of maximal hilar cell loss (when all hilar cells are removed) and the addition of a maximal number of previously nonexistent mossy fiber connections to other granule cells [the densest, anatomically quantified sprouting reported in the literature from an experimental epilepsy model was an average of 275 extra mossy fiber contacts per granule cell (Buckmaster et al. 2002b)—we considered this number 100% sprouting]. Therefore intermediate stages in the progression of sclerosis could be distributed between the control (0% sclerosis) and the maximally sclerotic (100% sclerosis) states. For example, at 50% sclerosis, 50% of mossy cells and 50% of hilar interneurons were lost, and 50% of the maximal sprouting of mossy fibers was implemented (Fig. 1B2). Sclerosis could also be studied in networks containing only the nodes representing the excitatory cells (“isolated excitatory graph”) or only the interneurons (“isolated inhibitory graph”). However, mossy fiber sprouting obviously could not be implemented in the isolated inhibitory graph. Similarly, sprouting could be studied without hilar cell loss (“sprouting-only networks”). However, the reverse was not necessarily true because mossy cell loss without mossy fiber sprouting in the isolated excitatory graph caused the graph to become disconnected as sclerosis progressed (because granule cells do not make synapses on each other in the control network). It should also be noted that in the isolated interneuronal graphs, axo-axonic cells were included only as synaptic targets for other interneurons, but not sampled for the L and C calculations, because they exclusively projected to excitatory neurons. In addition, the interneuronal graphs were characterized only Յ96.66% sclerosis because 100% sclerosis resulted in a disconnected graph.
DOM GRAPHS. Specific topological measures (the average path length and the clustering coefficient; see following text) were calculated for each structural model representing different stages in the progression of sclerosis, to quantify how network architecture changes during sclerosis. However, because the numbers of nodes and links change during sclerosis, these topological measures are meaningful only if they are contrasted with similar measures taken for equivalent random graphs at each stage of sclerosis. An

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FIG. 1. Schematic of the basic circuitry of the dentate gyrus and the changes to the network during sclerosis. A: relational representation of the healthy dentate gyrus illustrating the network connections between the 8 major cell types: GC, granule cell; BC, basket cell; MC, mossy cell; AAC, axo-axonic cells; MOPP, molecular layer interneurons with axons in perforant-path termination zone; HIPP, hilar interneurons with axons in perforant-path termination zone; HICAP, hilar interneurons with axons in the commissural/associational pathway termination zone; and IS, interneuron selective cells. Schematic shows the characteristic location of the various cell types within the 3 layers of the dentate gyrus. Note, however, that this diagram does not indicate the topography of axonal connectivity (present in both the structural and functional dentate models) or the somatodendritic location of the synapses (incorporated in the functional network models). B1: schematic of the excitatory connectivity of the healthy dentate gyrus is illustrated (only cell types in the hilus and granule cells are shown). Note that the granule cell axons (the mossy fibers) do not contact other granule cells in the healthy network. B2: schematic of the dentate gyrus at 50% sclerosis shows the loss (indicated by the large ✕ symbols) of half the population of all hilar cell types and the 50% sprouting of mossy fibers that results in abnormal connections between granule cells (note that, unlike in this simplified schematic, all granule cells formed sprouted contacts in the structural and functional models of sclerosis; thus progressive increase in sprouting was implemented by increasing the number of postsynaptic granule cells contacted by single sprouted mossy fibers; see METHODS). C: schematics of 3 basic network topologies: regular, small-world, and random. Nodes in a regular network are connected to their nearest neighbors, resulting in a high degree of local interconnectedness (high clustering coefficient C), but also requiring a large number of steps to reach other nodes in the network from a given starting point (high average path length L). Reconnection of even a few of the local connections in a regular network to distal nodes in a random manner results in the emergence of a small-world network, with a conserved high clustering coefficient (C) but a low average path length (L). In a random network, there is no spatial restriction on the connectivity of the individual nodes, resulting in a network with a low average path length L but also a low clustering coefficient C.
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TABLE 1. Connectivity matrix for the neuronal network of the control dentate gyrus

Granule Cells

Mossy Cells

Basket Cells

Axo-axonic Cells

MOPP Cells

HIPP Cells HICAP Cells

IS Cells

Granule cells (1,000,000) ref. [1–5]
Mossy cells (30,000) ref. [11]
Basket cells (10,000) ref. [16,17]
Axo-axonic cells (2,000) ref. [4,22]
MOPP cells (4,000) ref. [11,14]
HIPP cells (12,000) ref. [11]
HICAP cells (3,000) ref. [5,29,30]
IS cells (3,000) ref. [15,29,30]

X X ref. [6]
32,500 30,000–35,000 ref. [4,11–13]
1,250 1,000–1,500 ref. [4,16–19]
3,000 2,000–4,000 ref. [4,18,22]
7,500 5,000–10,000
ref. [14]
1,550 1,500–1,600 ref. [4,11,20]
700 700 ref. [4,11,20]
X X ref. [15]

9.5 7–12 ref. [7]
350 200–500 ref. [12,13]
75 50–100 ref. [11,16,17,19]
150 100–200 ref. [4,5,11,14,23]
X X ref. [14,24]
35 20–50 ref. [4,11,12,27,28]
35 30–40 ref. [20]
X X ref. [15]

15 10–20 ref. [6–9]
7.5 5–10 ref. [13]
35 20–50 ref. [16,17,20,21]
X X ref. [5,18]
40 30–50 ref. [14,25]
450 400–500 ref. [4,11,20]
175 150–200 ref. [4,11,20]
7.5 5–10 ref. [15,19]

3 1–5 ref. [6,7,9]
7.5 5–10 ref. [13]
X X ref. [18]
X X ref. [5,18]
1.5 1–2 ref. [14,26]
30 20–40 ref. [20,25]
X X ref. [20]
X X ref. [15]

X X ref. [6]
5 5 ref. [14]
X X ref. [18]
X X ref. [5,18]
7.5 5–10 ref. [14,25]
15 10–20 ref. [25]
15 10–20 ref. [14,20]

110 100–120 ref. [4,10,11]
600 600 ref. [12,13]
0.5 0–1 ref. [18]
X X ref. [5,18]
X X ref. [14,20,25]
X X ref. [14,20,25]
50 50 ref. [20]
7.5 5–10 ref. [19]

40 30–50 ref. [4,7,10,11]
200 200 ref. [12,13]
X X ref. [18]
X X ref. [5,18]
7.5 5–10 ref. [14,25]
15 10–20 ref. [25]
50 50 ref. [20]
7.5 5–10 ref. [19]

20 10–30 ref. [7]
X X ref. [15]
X X ref. [10,20]
X X ref. [5,18,19]
X X ref. [14,15]
X X ref. [15,20]
450 100–800 ref. [15]

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Cell numbers and connectivity values were estimated from published data for granule cells, Mossy cells, basket cells, axo-axonic cells, molecular layer
interneurons with axons in perforant-path termination zone (MOPP), hilar interneurons with axons in perforant-path termination zone (HIPP), hilar interneurons
with axons in the commissural/associational pathway termination zone (HICAP), and interneuron-selective cells (IS). Connectivity is given as the number of
connections to a postsynaptic population (row 1) from a single presynaptic neuron (column 1). The average number of connections used in the graph theoretical
calculations is given in bold. Note, however, that the small-world structure was preserved even if only the extreme low or the extreme high estimates were used for the calculation of L and C (for further details, see APPENDIX B1(3). References given correspond to: 1Gaarskjaer (1978); 2Boss et al. (1985); 3West (1990); 4Patton and McNaughton (1995); 5Freund and Buzsa´ki (1996); 6Buckmaster and Dudek (1999); 7Acsa´dy et al. (1998); 8Geiger et al. (1997); 9Blasco-Ibanez et al. (2000); 10Gulya´s et al. (1992); 11Buckmaster and Jongen-Relo (1999); 12Buckmaster et al. (1996); 13Wenzel et al. (1997); 14Han et al. (1993); 15Gulya´s et al. (1996); 16Babb et al. (1988); 17Woodson et al. (1989); 18Halasy and Somogyi (1993); 19Acsa´dy et al. (2000); 20Sik et al. (1997); 21Bartos et al. (2001); 22Li et al. (1992); 23Ribak et al. (1985); 24Frotscher et al. (1991); 25Katona et al. (1999); 26Soriano et al. (1990); 27Claiborne et al. (1990); 28Buckmaster et al. (2002a); 29Nomura et al. (1997a); 30Nomura et al. (1997b).

equivalent random graph has the same numbers of nodes and links as the graph (representing a particular degree of sclerosis) to which it is compared, although the nodes have no representation of distinct cell types and possess uniform connection probabilities for all nodes. For example, the equivalent random graph for the control (0% sclerosis) structural model has about a million nodes and the same number of links as in the control structural model, but the nodes are uniform (i.e., there is no “granule cell node,” as in the structural model) and the links are randomly and uniformly distributed between the nodes.
CONSTRUCTION OF THE FUNCTIONAL MODEL. The effects of structural changes on network excitability were determined using a realistic functional model of the dentate gyrus (note that “functional” refers to the fact that neurons in this model network can fire spikes, receive synaptic inputs, and the network can exhibit ensemble activities, e.g., traveling waves; in contrast, the structural model has nodes that exhibit no activity). The functional model contained biophysically realistic, multicompartmental single-cell models of excitatory and inhibitory neurons connected by weighted synapses, as published previously (Santhakumar et al. 2005). Unlike the structural model, which contained eight cell types, the functional model had only four cell types, as a result of the insufficient electrophysiological data for simulating the other four cell types. The four cell types that were in the functional model were the two excitatory cells (i.e., the granule cells and the mossy cells) and two types of interneurons (the somatically projecting fast spiking basket cells

and the dendritically projecting HIPP cells; note that these represent two major, numerically dominant, and functionally important classes of dentate interneurons, corresponding to parvalbumin- and somatostatin-positive interneurons; as indicated in Table 1, basket cells and HIPP cells together outnumber the other four interneuronal classes by about 2:1). Because the functional model had a smaller proportion of interneurons than the biological dentate gyrus, control simulations (involving the doubling of all inhibitory conductances in the network) were carried out to verify that the observed changes in network excitability during sclerosis did not arise from decreased inhibition in the network, i.e., that the conclusions were robust (see RESULTS and APPENDIX B3).
Although the functional model was large, because of computational limitations, it still contained fewer neurons (a total of about 50,000 multicompartmental model cells) than the biological dentate gyrus (about one million neurons) or the full-scale structural model (about one million nodes). Because of this 1:20 reduction in size, a number of measures had to be taken before examining the role of structural changes on network activity. First, we had to build a structural model of the functional model itself (i.e., a graph with roughly 50,000 nodes) and verify that the characteristic changes in network architecture observed in the full-scale structural model of the dentate gyrus occur in the 1:20 scale structural model (graph) of the functional model as well. Second, certain synaptic connection strengths had to be adjusted from the experimentally observed values (see following text).

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FIG. 2. Gaussian fits to experimentally determined distributions of axonal branch length used in construction of the models of the dentate gyrus. A: plot shows the averaged axonal distribution of 13 granule cells (Buckmaster and Dudek 1999) and the corresponding Gaussian fit. B: fit to the septotemporal distribution of axonal lengths of a filled and reconstructed basket cell (Sik et al. 1997). C: fit to the axonal distribution of a CA1 axo-axonic cell (Li et al. 1992). D: Gaussian fit to the averaged axonal distributions of 3 HIPP cells from gerbil (Buckmaster et al. 2002a). E: fit to averaged axonal distributions of 3 mossy cells illustrates the characteristic bimodal pattern of distribution (Buckmaster et al. 1996). F: histogram of the axonal lengths of a HICAP cell along the long axis of the dentate gyrus (Sik et al. 1997) and the Gaussian fit to the distribution. All distributions were based on axonal reconstruction of cells filled in vivo. In all plots, the septal end of the dentate gyrus is on the left (indicated by negative coordinates) and the soma is located at zero.

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Implementation and assessment of the models
IMPLEMENTATION OF THE STRUCTURAL MODEL. The dentate gyrus was represented as a 6-mm strip (corresponding to the approximate septotemporal extent of the rat dentate gyrus; West et al. 1978) subdivided into 60-␮m bins. Cells of the eight distinct neuronal types were represented in the structural model as individual nodes and distributed evenly among the bins. The nodes were linked according to cell-type–specific connection probabilities derived from the average number of projections from the pre- to the postsynaptic neuronal class in the literature (i.e., according to the connectivity matrix shown in Table 1; appendixes A1 and A2). In general, in addition to the mere existence of connections between two particular cell types (codified in Table 1), the probability of connections from one particular cell A to a given cell B also depends on the extent of the axonal arbor of cell A and the relative distance between cells A and B. Therefore the cell-type–specific connection probability was further modified by a factor obtained by the normalized Gaussian fits to the experimentally determined axonal distributions of the presynaptic cells (APPENDIX A3 and Fig. 2) and the relative positions of the pre- and postsynaptic neurons in the graph. Within these cell-type–specific constraints, connections were made probabilistically on a neuron to neuron (or, more specifically, because we are talking about a graph, a “node to node”) basis with a uniform synapse (“outgoing link”) density along the axon [in agreement with the in vivo data in Sik et al. (1997)], treating multiple synapses between two cells as a single link and excluding autapses. Note that this implementation of the structural model did not take into account certain potential factors that may distort local connection probabilities (see DISCUSSION). Also note that because the neuronal origin of GABAergic sprouting is unknown (Andre et al. 2001; Esclapez and Houser 1999), only sprouting of mossy fiber connections were included in sclerotic graphs.
CHARACTERISTICS. To quantify the topological characteristics of the structural model, the approach of Watts and Strogatz (1998),

originally applied to the neuronal network of the worm C. elegans, was used. Two measures were used to assess the salient features of the structural models: the average path length L (average number of steps to reach any node in the network) reflecting global connectivity and the average clustering coefficient C (for a given node, the fraction of possible connections between its postsynaptic nodes that actually exist) as a measure of local connectivity. In human societies, for example, C describes the probability that friends of person X also know each other (i.e., it is a measure of local “cliquishness”), whereas L describes what is commonly known as “the six degrees of separation” between any two persons on the planet (i.e., it is a measure of large-distance or “global” connectivity). These two key topological measures for the structural model of the dentate gyrus were calculated using custom C code on a Tyan Thunder 2.0 GHz dual Opteron server (32 GB RAM). Graph calculation times were roughly 50 h per graph.
In general, there are three distinct major network topologies (for reviews, see Buzsa´ki et al. 2004; Soltesz 2006): 1) Regular (high L, high C); 2) Random (low L, low C); and 3) Small world (low L, high C) (Fig. 1C). The graph of a regular (or “ordered” or “lattice-like”) network is characterized by a high degree of local interconnectedness (because each node is linked to its nearest neighbors, resulting in a high C), but nodes at the two ends of the graph are separated by a large number of nodes (leading to a high L). In other words, a regular network has an abundance of local connections (thus the comparison to a “lattice” or a fishing net), but has no long-distance connections. Conversely, the graph of a random network is well connected globally (low L), but its local connectivity is low (low C) (this is because random connectivity does not typically form local clusters, but it results in numerous long-distance connections). A small-world structure can be best understood by considering that it can be derived from a regular network by disconnecting and randomly reconnecting a few of its connections (leading to at least a few long-distance connections, which, in turn, results in a low L while retaining the high C of the regular network) (Fig. 1C; note that the term “random reconnection” is used here for didactic purposes to describe a commonly used

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derivation of a small-world network from a regular network, without implying that long-distance connections in an actual biological smallworld network are random). Therefore small-world networks are both locally (high C) and globally (low L) well connected (again, in the language of social networks, humans tend to have a strongly interconnected local cluster of friends, but also at least a few acquaintances with connections far outside of the local circle). The quantitative determination of the small-world topology of a given network is performed by comparison to an equivalent random graph: for a small-world network, L Ϸ Lrandom and C ϾϾ Crandom.
Given the large size of the graph, L and C were determined from the weighted averages of randomly sampled nodes (“weighted” here refers to the fact that our sampling took into account the ratio of the nodes representing granule cells, mossy cells, and the six interneuronal classes; i.e., the sampling of the nodes in the structural model had to reflect the ratio of the constituent cell types). A minimum of 1/1,000 granule and mossy cells and 1/100 interneurons were sampled and control calculations were performed to verify the accuracy and stability of the sampling method, as described in APPENDIX B1.
IMPLEMENTATION OF THE FUNCTIONAL MODEL. The functional model network was implemented using the NEURON 5.6 simulation environment (Hines and Carnevale 1997). The required simulation times ranged from about 35 to 70 h per model network. The single-cell models were taken from Santhakumar et al. (2005), with morphologies, voltage-gated conductances, and intrinsic properties based on detailed experimental data. Briefly, the single-cell models had nine to 17 compartments including a somatic compartment and two to four dendrites. Minimally, each dendrite was modeled with a proximal, middle, and distal dendritic segment. The models contained nine classes of active conductance mechanisms such as sodium channels, three types of potassium channels (A-type and fast and slow delayed rectifier), three types of calcium channels (L-, N-, and T-type), two types of calcium-dependent potassium channels (SK and BK channels), Ih, and an intracellular calcium clearance process. The intrinsic properties of the cell types were modeled to simulate the passive (membrane potential at rest, input resistance, and membrane time constant) and active (amplitude and threshold of action potential, fast afterhyperpolarization, spike frequency adaptation, and sag ratios) properties observed in experimental data (Lubke et al. 1998; Staley et

al. 1992). For granule cells, the somatodendritic distribution of active conductances was adapted from Aradi and Holmes (1999). In all other cell types, the active conductances, with the exception of sodium and fast delayed rectifier potassium channels, were distributed uniformly in all compartments. Sodium and fast delayed rectifier potassium conductances were present only in the soma and proximal dendritic compartments. Additionally, correction for the membrane area contribution of spines was implemented for the granule and mossy cell models. The multicompartmental single-cell models of 50,000 granule, 1,500 mossy, 500 basket, and 600 HIPP cells were evenly distributed in 100 bins along the septotemporal axis.
Connectivity in the functional model network was established using the procedure described for the structural model. All connection probabilities were increased fivefold compared with the structural model, to compensate for the fewer number of cells in the functional model and ensure that no cells in the model networks were disconnected (note that even with this increase in connection probability, each presynaptic cell still made fewer connections in the functional model network than in the full-scale structural model because the postsynaptic cell populations were reduced by a factor of 20; compare Tables 1 and 2). The synaptic conductances between cell types, based on unitary conductances from the literature, were taken from Santhakumar et al. (2005). Excitatory synaptic conductances were adjusted to avoid depolarization block in postsynaptic cells arising from the higher value of the clustering coefficient C in the functional model network (see Fig. 3B). Except when specifically stated (see APPENDIX B3), distance-dependent axonal conduction delays were not included. Perforant path stimulation was simulated as in Santhakumar et al. (2005), by a single synaptic input to 5,000 granule cells, 10 mossy cells (note that only a fraction of all mossy cells receive direct perforant path input; Buckmaster et al. 1992; Scharfman 1991), and 50 basket cells (situated in the middle lamella of the model network) at t ϭ 5 ms after the start of the simulation. Additional details of the functional model network, including the convergence and divergence of the connections and the synaptic weights, are listed in Table 2. Note that the current functional model has three primary differences from the network model of Santhakumar et al. (2005). First, we have enlarged the network by two orders of magnitude, making it possible to study the small-world network characteristics of the dentate gyrus.

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TABLE 2. Parameters of functional network model

Granule cells* (50,000)
Mossy cells (1,500)
Basket cells (500)
HIPP cells (600)
Perforant path†

To 3
Convergence Divergence Synapse weight, nS Convergence Divergence Synapse weight, nS Convergence Divergence Synapse weight, nS Convergence Divergence Synapse weight, nS Synapse weight, nS

Functional Model Network Parameters



68.03 68.03 1.00 243.62 8,120.82 0.30
3.11 313.22
1.60 4.82 401.86 0.50 20.00

78.05 2.34 0.20 87.23 87.23 0.50 6.31 18.93 1.50 3.76 9.39 1.00 17.50

370.95 3.71 0.94 5.59 1.86 0.30 8.98 8.98 7.60
140.13 116.77
0.50 10.00

2,266.64 27.19 0.10 375.53 150.21 0.20 n/a n/a n/a n/a n/a n/a n/a

The cell numbers (column 1) and synaptic connectivity values and strengths in the functional model network are used for the activity calculations in Fig. 4 (quantified in Fig. 5). Note that this network is smaller (50,000ϩ cells) than the full-scale dentate gyrus (Ͼ1,000,000 cells); thus the connectivity had to be adjusted from what is shown in Table 1. Convergence is given as the number of connections converging onto a single postsynaptic neuron (row 1) from a presynaptic neuronal population (column 1). For example, 243 mossy cells converge on a single granule cell in this network. Divergence is given as the number of connections diverging to a postsynaptic population (row 1) from a single presynaptic neuron (column 1). For example, a single mossy cell makes synapses on 8,120 postsynaptic granule cells in this network. The strengths of the connections are given in nanosiemens (nS). For example, the strength of the excitatory synapse formed by a single mossy cell on a single granule cell is 0.3 nS. *Granule cell to granule cell connections represent values at 100% sprouting. †Perforant path input to 5,000 granule cells (two synapses each), 50 basket cells (two synapses each), and 10 mossy cells (one synapse each) in the central 10 bins of the network model.

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Second, in contrast to the Santhakumar et al. (2005) study that focused on moderate (Ͻ50%) sclerosis, the current model examines the structural and functional effects of the progression of sclerosis from 0 to 100%. Third, the use of Gaussian fits to constrain axonal distributions instead of an uniform probability adopted in the earlier model considerably increased topological accuracy of this model. Moreover, the current study also tested the effects of hilar interneuronal loss and parallel increases in sprouting and hilar cell loss in contrast to the independent examination of sprouting and mossy cell loss performed in Santhakumar et al. (2005).
CITABILITY. Excitability of the functional model was assessed by a number of measures, including: 1) total duration of the granule cell discharges in the network (defined as the time from the first spike fired by a granule cell in the network to the last spike fired by a granule cell in the network; note that the first and the last granule cell spikes may originate from different granule cells); 2) mean number of spikes per granule cell; 3) latency to spread of activity from the perforant path activation to the firing of the most distant granule cells in the network; and 4) synchrony of granule cell discharges. Because the latter measure is the most complicated, it will be described below separately.
To assess synchrony, the coherence of granule cell firing between 100 and 200 ms (i.e., sufficiently far in time from the initial stimulus, and during a period where networkwide activity could be observed at most degrees of sclerosis; see Fig. 4) was calculated, using a published coherence measure (Foldy et al. 2004; White et al. 1998). The local coherence was calculated by all-to-all comparison of the activity in granule cells #25000 to #25999. Pairwise comparison of the activity in granule cells #25000 to #25999 and #45000 to #45999 provided the long range coherence. To calculate coherence from the network simulations during the postsimulation analysis, trains of square pulses were generated for each firing cell in a pair with each pulse of unitary height centered on the spike peak and the width equal to 20% of the mean interspike interval of the faster spiking cell in the pair. Subsequently, the shared area of the unit height pulse trains was calculated (equivalent to the zero time lag cross-correlation). Coherence was

FIG. 3. Alterations in L and C with sclerosis for the various structural models of the dentate gyrus. A, C, and E: changes in average path length L with sclerosis. B, D, and F: changes in clustering coefficient C with sclerosis. Explanation of symbols in A also applies to B–D. Explanation of symbols in E also applies to F. Black lines in A–D: full-scale structural models. Black lines in E and F: isolated excitatory/inhibitory graphs. Blue lines: structural model of the functional model with sclerosis. Green lines: structural model of the functional model network with sprouting only. Dashed lines in A and B: equivalent random graphs of the full-scale structural model. A and B: plots of the changes in L and C of the various dentate graphs. L and C for the full-scale structural model of the healthy (i.e., at 0% sclerosis) dentate gyrus are marked with “E” on the y-axis. C and D: plots for relative L (ϭL/Lrandom, from A) and relative C (ϭC/Crandom, from B). In C and D, dotted horizontal lines indicate the relative L and C for the full-scale structural model of the healthy dentate graph; vertical dotted lines indicate the degree of sclerosis where the relative L exceeds and the relative C decreases below the values for the control graph. Note the close similarity of the relative L and C changes during sclerosis in the structural model of the functional model network (50,000ϩ nodes; blue lines) and in the full-scale structural model (Ͼ1 million nodes; solid black lines). E and F: plots of changes in L and C for the isolated excitatory and inhibitory graphs with sclerosis and for the isolated excitatory graphs with sprouting alone (without mossy cell loss), respectively. Changes in L and C for inhibitory interneurons after hilar interneuron loss: dotted lines; for excitatory cell types (granule cells and mossy cells): solid lines; for mossy fiber sprouting in the absence of concurrent mossy cell loss: dashed lines. Note the 2 y-axes in B and F.
defined as the sum of their shared areas normalized by the square root of the product of the total areas of the individual trains (Foldy et al. 2004; White et al. 1998).
Data analysis and plotting were done using Matlab 6.5.1 (The MathWorks, Natick MA) and Sigmaplot 8.0 (SPSS, Chicago IL).
Note that the structural and functional model networks are available for download from ModelDB (http://senselab.med.yale. edu/senselab/ModelDB).
Key features of the biological net captured by the structural model
The structural model of the healthy, nonsclerotic dentate gyrus contained over one million (1,064,000) nodes, with the majority (94%) representing granule cells. The million nodes in the control dentate graph were richly linked by over a billion links (1,287,363,500). As in the biological network, there was a large difference in the degree of interconnectedness between nodes representing different cell types. The nodes representing granule cells gave the fewest links (Table 1) and these links were also the most spatially restricted (corresponding to the restricted septotemporal extent of the in vivo filled granule cell axons shown; Figs. 1B and 2). In contrast, nodes representing mossy cells formed by far the highest number of links to other nodes (Table 1) and these links spanned almost the entire length of the dentate graph (corresponding to the large extent of single mossy cell axon arbors; Figs. 1B and 2).
Globally and locally well connected nature of the control dentate gyrus
We assessed the quantitative topological properties of the control structural model of the dentate gyrus by calculating L and C for the graph at 0% sclerosis and for the equivalent

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FIG. 4. Effects of the sclerosis-related topological changes on granule cell activity in functional model networks. A–F: raster plots of the first 300 ms of action potential discharges of granule cells in the functional model network (Granule cells #1 to #50,000, plotted on the y-axis) at increasing degrees of sclerosis. Network activity was initiated by a single stimulation of the perforant path input to granule cells #22,500 to #27,499 and to 10 mossy cells and 50 basket cells (distributed in the same area as the stimulated granule cells) at t ϭ 5 ms (as in Santhakumar et al. 2005). Perforant path activation led to an initial spike in the directly stimulated granule cells (vertically aligned dots at t ϭ 14 ms), followed by a gap in granule cell activity resulting from inhibition by local, directly stimulated basket cells. Note that the most pronounced hyperactivity was observed at submaximal (80%) sclerosis (for quantification, see Fig. 5, A–D).

random graph. The average path length for the control dentate graph was remarkably low (L ϭ 2.68, marked E in Fig. 3A), considering the presence of over one million nodes in the network. The L ϭ 2.68 value indicated that, on average, fewer than three synapses separated any two neurons in the dentate gyrus. Therefore the low L showed that the graph was well connected globally. To our knowledge, this is the first measurement of L for a mammalian microcircuit, where each neuron is represented by a unique node in the graph. It is interesting to note that the average path length for the control dentate graph was virtually identical to the L ϭ 2.65 reported for the much smaller nervous system of the worm C. elegans with a connected graph of only 282 nodes (Watts and Strogatz 1998) (note, however, that the C. elegans simulations were done on a nondirected graph, whereas our graphs take into account the directionality of the connections).
The average path length calculated for the equivalent random graph was only slightly lower (Lrandom ϭ 2.25) than the L for the control structural model, resulting in a L/Lrandom ratio close to one (1.19; indicated by the solid black line at 0% sclerosis in Fig. 3C). However, the control structural model was much more highly connected locally than the equivalent random graph, as shown by the high value of the relative clustering coefficient (C/Crandom ϭ 0.026751 / 0.001135 ϭ

24.7, indicated by the solid black line at 0% sclerosis in Fig. 3D; note that the control value for C is marked E in Fig. 3B). The relatively low average path length and high clustering coefficient of the control dentate graph fulfilled the dual requirements of L Ϸ Lrandom and C ϾϾ Crandom, demonstrating that the normal, healthy biological dentate gyrus is a smallworld network (Watts and Strogatz 1998).
Enhanced local and global connectivity with submaximal sclerosis and the transition to a more regular network structure at severe sclerosis
Next, we determined how the graph characteristics of the dentate gyrus change during the progression of sclerosis, characterized by the loss of hilar neurons and mossy fiber sprouting (Longo et al. 2003; Nadler 2003; Ratzliff et al. 2004). The fully sclerotic dentate graph exhibited only a small (4.5%) decrease in the total number of nodes (48,000 nodes representing hilar cells lost out of 1,064,000), but there was a dramatic (74%) reduction in the number of links (953,198,800 links removed out of the total 1,287,363,500), indicating that maximal sprouting did not replace the lost links resulting from the removal of all richly connected hilar cells (Table 1). To determine how small-world topology was affected by the removal of so many links, L and C values were calculated for dentate graphs

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constructed at various degrees of sclerosis (solid black lines in Fig. 3, A and B). Because the average path length and clustering coefficient of the equivalent random graphs also changed with deletion of nodes and addition of links, these were recalculated for each degree of sclerosis (dashed lines in Fig. 3, A and B).
The results revealed that the progression of sclerosis (increasing loss of the predominantly long distance projecting mossy cells and hilar interneurons, with increasing degrees of spatially restricted mossy fiber sprouting) did not significantly elevate L, until about 90% of the hilar cell nodes were lost (solid black line in Fig. 3A). In fact, the relative average path length (L/Lrandom) actually decreased below the control, 0% sclerotic level (illustrated by the horizontal dotted line in Fig. 3C). These data indicated that, despite the loss of long-distance projecting hilar cells and the resulting massive decrease in connections, there was a seemingly paradoxical enhancement of long-distance connectivity and conservation of the smallworld topology during submaximal (Ͻ90%) sclerosis. However, the relative L did not stay below its control value because it showed a sudden increase during the last stages of sclerosis. Therefore the changes in relative L during sclerosis were strongly biphasic (i.e., the initial decrease in L was followed by a sudden increase). Importantly, as illustrated in Fig. 3C, it was only at 96.6% sclerosis (vertical dotted line) that the relative average path length (solid black line) started to increase above the control value (horizontal dotted line), indicating that global connectivity was preserved until the final stages of sclerosis.
The high C value (the second characteristic feature of small-world topology) of the control dentate graph was also preserved and actually enhanced during submaximal sclerosis. Although the initial increase in C values was followed by a subsequent decrease at midlevel sclerosis (around 40%) (solid black line in Fig. 3B), the relative clustering coefficient (C/Crandom, solid black line in Fig. 3D) increased above the control value (indicated by horizontal dotted line in Fig. 3D) up to about 90% sclerosis, showing a sclerosis-related enhancement of local connectivity. Similar to the biphasic changes in relative L, it was only shortly before the onset of full sclerosis that the relative C values decreased below the control level (dotted lines in Fig. 3D; note that, even though relative clustering coefficient decreased at 100% sclerosis in Fig. 3D, the absolute clustering coefficient in Fig. 3B remained more than tenfold higher than Crandom even at maximal sclerosis).
The decreasing relative average path length (solid black line in Fig. 3C) and increasing relative clustering coefficient (solid black line in Fig. 3D) during submaximal sclerosis together demonstrated an unexpected enhancement of the features characterizing a small-world topology. However, a transition to a more regular or lattice-like network structure (Watts and Strogatz 1998), characterized by high values of both L and C (i.e., poor global but rich local connectivity), occurred shortly before maximal (100%) sclerosis (note that the fully sclerotic network is not a true lattice structure with only nearestneighbor connections because, e.g., the axonal arbors of basket cells span roughly 25% of the septotemporal extent of dentate gyrus, providing a large number of midrange connections in the network even at 100% sclerosis).

Analysis of the changing roles of topological factors using
isolated excitatory and inhibitory structural models
To determine the mechanisms underlying the transient enhancement of small-world properties during sclerosis, graphs of the excitatory and inhibitory parts of the dentate network were considered separately. First, the isolated excitatory graph was examined. The sclerosis-induced changes in L and C in the isolated excitatory graph (mossy cells and granule cells alone; solid lines in Fig. 3, E and F) were generally similar to the alterations in the full dentate graph (solid black lines in Fig. 3, A and B), suggesting that it was the loss of long-range connections (in this case, from mossy cells) and mossy fiber sprouting that played key roles in alterations of graph structure. Sprouting without mossy cell loss did not significantly affect the average path length in the excitatory graph (dashed line in Fig. 3E), indicating that the added local connections from sprouted mossy fibers mattered little for L when the long-range connections of the mossy cells were retained. However, sprouting without mossy cell loss in the excitatory graph produced similar changes in the clustering coefficient (dashed line in Fig. 3F) as sprouting with mossy cell loss (solid line in Fig. 3F). (Note that the essentially unchanged L and the biphasic changes in C observed in the sprouting-only isolated excitatory network will play an important role in determining the role of L and C in network hyperexcitability during sclerosis; see following text.) The decrease in C at higher degrees of sclerosis (which is also observed in the structural model containing both excitatory and inhibitory neurons) was the result of each granule cell primarily contacting other granule cells after mossy fiber sprouting. Because the probability of sprouted connections between any two granule cells is low, the fraction of actually existing connections between pairs of postsynaptic granule cells is also low, resulting in a decreasing C (see METHODS, ASSESSMENT OF THE STRUCTURAL MODEL: CALCULATION OF GRAPH CHARACTERISTICS). In the structural model network containing both excitatory and inhibitory neurons, this dominant influence of granule-to-granule cell connections on the clustering coefficient was more gradual as a result of the larger number of nongranule cell postsynaptic targets of each granule cell.
In contrast to the isolated excitatory graph, the isolated interneuronal graph (i.e., without granule cells and mossy cells) showed a steady increase in average path length and decrease in clustering coefficient with sclerosis (dotted lines in Fig. 3, E and F) because the progressive loss of hilar interneurons resulted in an increasingly sparse graph (note that there were no granule cells and thus no sprouting of mossy fibers in the isolated interneuronal graph). Interestingly, the control interneuronal graph had an order of magnitude higher clustering coefficient (C ϭ 0.0561) than the control excitatory graph (C ϭ 0.0060), reflecting the significantly more interconnected nature of interneuronal circuits.
These results showed that, during submaximal sclerosis, it was primarily the sprouting of mossy fibers that played a key role in determining topology because the local shortcuts provided by sprouting not only increased C, but also maintained a low L: for granule cells (GC), the loss of mossy cells (MC) removed a number of two-step (GC 3 MC 3 GC) and three-step (GC 3 MC 3 MC 3 GC) paths that were partially compensated by the introduction of a large number of new

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ModelSclerosisFigNetworkDentate Gyrus