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Topological Determinants of Epileptogenesis in Large-Scale Structural and Functional Models of the Dentate Gyrus Derived From Experimental Data

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

	ABSTRACT

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX A1: ESTIMATION OF... APPENDIX A2: ESTIMATION OF... APPENDIX A3: FITS TO... APPENDIX B1: CONTROLS FOR... APPENDIX B2: CONTROLS FOR... APPENDIX B3: CONTROLS FOR... GRANTS ACKNOWLEDGMENTS REFERENCES

 
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 single-cell 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.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX A1: ESTIMATION OF... APPENDIX A2: ESTIMATION OF... APPENDIX A3: FITS TO... APPENDIX B1: CONTROLS FOR... APPENDIX B2: CONTROLS FOR... APPENDIX B3: CONTROLS FOR... GRANTS ACKNOWLEDGMENTS REFERENCES

 
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 seizure-induced end-folium sclerosis (Margerison and Corsellis 1966; Mathern et al. 1996), indicating the emergence of a fundamentally transformed microcircuit. Because structural alterations in 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 (Buzsáki 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; Barabási 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 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.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX A1: ESTIMATION OF... APPENDIX A2: ESTIMATION OF... APPENDIX A3: FITS TO... APPENDIX B1: CONTROLS FOR... APPENDIX B2: CONTROLS FOR... APPENDIX B3: CONTROLS FOR... GRANTS ACKNOWLEDGMENTS REFERENCES

 
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.

View larger version (32K): [in this window] [in a new window]   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 x 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.

View larger version (43K): [in this window] [in a new window]   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.

CONSTRUCTION OF THE STRUCTURAL MODEL—PROGRESSIVE SCLEROSIS. 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 µm) 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 96.66% sclerosis because 100% sclerosis resulted in a disconnected graph.

CONSTRUCTION OF THE STRUCTURAL MODEL—EQUIVALENT RANDOM 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 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).

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-µm 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.

ASSESSMENT OF THE STRUCTURAL MODEL: CALCULATION OF GRAPH 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 Buzsáki 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 derivation of a small-world network from a regular network, without implying that long-distance connections in an actual biological small-world 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 L_random and C >> C_random.

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), I_h, 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 = 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. Second, in contrast to the Santhakumar et al. (2005) study that focused on moderate (<50%) 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).

View larger version (32K): [in this window] [in a new window]   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 "" on the y-axis. C and D: plots for relative L (=L/Lrandom, from A) and relative C (=C/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 (50,000+ nodes; blue lines) and in the full-scale structural model (>1 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.

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 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).

View larger version (35K): [in this window] [in a new window]   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 = 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 = 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).

Note that the structural and functional model networks are available for download from ModelDB (http://senselab.med.yale.edu/senselab/ModelDB).

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX A1: ESTIMATION OF... APPENDIX A2: ESTIMATION OF... APPENDIX A3: FITS TO... APPENDIX B1: CONTROLS FOR... APPENDIX B2: CONTROLS FOR... APPENDIX B3: CONTROLS FOR... GRANTS ACKNOWLEDGMENTS REFERENCES

 
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 random graph. The average path length for the control dentate graph was remarkably low (L = 2.68, marked in Fig. 3A), considering the presence of over one million nodes in the network. The L = 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 = 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 (L_random = 2.25) than the L for the control structural model, resulting in a L/L_random 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/C_random = 0.026751 / 0.001135 = 24.7, indicated by the solid black line at 0% sclerosis in Fig. 3D; note that the control value for C is marked in Fig. 3B). The relatively low average path length and high clustering coefficient of the control dentate graph fulfilled the dual requirements of L L_random and C >> C_random, demonstrating that the normal, healthy biological dentate gyrus is a small-world 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 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/L_random) 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 small-world topology during submaximal (<90%) 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/C_random, 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 C_random 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 nearest-neighbor 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 = 0.0561) than the control excitatory graph (C = 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 MC GC) and three-step (GC MC MC GC) paths that were partially compensated by the introduction of a large number of new three-step paths (GC MC GC GC; GC GC MC GC). This mechanism was similar for connections involving other long-range projecting hilar neurons. However, with severe sclerosis, the primary role shifted to the loss of hilar cells because their numbers became too small to maintain the low average path length (even with maximal sprouting). Therefore these calculations, carried out in the isolated excitatory and inhibitory networks, revealed that mossy fiber sprouting was primarily responsible for the transient increase in C at submaximal sclerosis, whereas the dramatic increase in L at maximal sclerosis was mostly explained by the loss of the last distantly projecting hilar neurons. In other words, the key topological determinant switches from sprouting to hilar cell loss during severe sclerosis.

Biphasic alterations in network topology during sclerosis result in biphasic changes in network excitability

Next, the functional consequences of the biphasic alterations in network topology observed in the structural model were explored in our large-scale, topographically, and biophysically realistic functional model of the dentate gyrus (for details on the model, see METHODS). Before the simulations, we first verified that the changes in L and C for the graph of the functional model network with 50,000+ cells were similar to those observed for the graph of the biological network containing 1,000,000+ cells (blue lines in Fig. 3, A–D; compare with solid black lines in the same panels).

As with the biological dentate gyrus (Santhakumar et al. 2001), the functional model network of the normal, nonsclerotic dentate gyrus showed only limited firing in response to single simulated perforant path stimulation (Fig. 4A). Between 20 and 80% sclerosis (when the features characteristic of small-world topology were found to be enhanced in the structural model), the functional model network showed increasing hyperexcitability (Fig. 4, B–E), with activity spreading to the entire network by 40% sclerosis, reminiscent of the epileptiform activity recorded in vitro (Rafiq et al. 1995). However, beyond 80% sclerosis, corresponding to the change from a small-world network to a more regular network in the structural model, the activity in the functional model network decreased (Fig. 4F). In other words, changes in network activity during sclerosis, as assessed by a probing stimulus to the perforant path, appeared to be biphasic, similar to the biphasic nature of the topological alterations during sclerosis observed in the structural model. It is interesting to note that the structural changes also affected network dynamics, as indicated by the transition from a spatially relatively uniform pattern (40–60% sclerosis; Fig. 4, C and D) to a pattern with distinct waves of activity (80–100% sclerosis; Fig. 4, E and F), that could collide and mutually annihilate (Netoff et al. 2004; Roxin et al. 2004).

Next, we quantified the changes in activity seen in Fig. 4 in the functional model network during sclerosis, using a number of measures of hyperexcitability (see METHODS). Both the maximal duration of granule cell activity and the mean number of spikes fired per granule cell (black lines in Fig. 5, A and B) increased and peaked at 80% sclerosis, followed by a decrease at 100% sclerosis. For degrees of sclerosis where the activity spread to the entire functional model network (40–100% sclerosis), the latency from perforant path activation to the firing of the most distant granule cells (black line in Fig. 5C) was the shortest (i.e., activity spreads the fastest) at 60% sclerosis (89.7 ms), followed by an increase in latency to the maximal value at 100% sclerosis (106.2 ms). The topological alterations also affected synchrony of firing in the network. Comparison of local and global synchrony in granule cell firing showed that local and long-range coherences initially were similar at 40 and 60% sclerosis (Fig. 5D, dashed black line: local coherence; solid black line: long-range coherence), but the long-range coherence subsequently decreased with increasing sclerosis, whereas the local coherence dramatically increased. This divergence of local and long-range synchrony with progressive sclerosis reflected the increasing dominance of short-range connections over long-range hilar cell axonal projections. Taken together, these measures all indicated that biphasic changes in the structural model resulted in biphasic alterations in hyperexcitability during sclerosis.

View larger version (30K): [in this window] [in a new window]   FIG. 5. Biphasic changes in granule cell firing in the functional model of the dentate gyrus during progressive sclerosis and the lack of biphasic changes when sprouting occurs without hilar neuron loss. Black lines: sclerosis; green lines: sprouting alone, without concurrent hilar cell loss. A: changes in the maximal duration of perforant path stimulation-evoked granule cell firing (for a description of this measure, see METHODS) as a function of sclerosis or sprouting alone (note that granule cell firing persisted beyond 2,000 ms at 80 and 100% sprouting in the sprouting-only case). B: mean number of spikes fired per granule cell with progressive sclerosis or sprouting without cell loss. C: latency to firing of granule cells in all 100 bins, in networks with persistent activity in all granule cell bins. D: local (dashed line, all-to-all comparison of action potential firing in granule cells #25000 to #25999) and long range (solid line, pairwise comparison of the activity in granule cells #25000 to #25999 to granule cells #45000 to #45999) coherence of granule cell firing in the time interval of 100 to 200 ms for 40–100% sclerosis or sprouting without cell loss. Note that in C and D, results are shown only for 40–100% sclerosis or sprouting, where activity spread to the entire network and persisted beyond 100 ms. E and F: changes in average total peak conductance of excitatory (E) and inhibitory (F) synapses onto the model granule cell during sclerosis or sprouting in the absence of cell loss. Note the lack of biphasic changes in A–D when sprouting took place without hilar cell loss.

Network activity parallels changes in sprouting in the absence of cell loss

The described biphasic changes in network hyperexcitability during sclerosis (solid black lines in Fig. 5, A–D) occurred in conjunction with biphasic changes in both relative average path length and relative clustering coefficient (solid black and blue lines in Fig. 3, C and D). Which of these two topological measures primarily determine changes in network hyperexcitability? How do the changes in L and C correspond to cell loss and mossy fiber sprouting? To answer these questions, we took advantage of the fact that (as noted before), in graphs of the isolated excitatory network with sprouting but no cell loss (dashed lines in Fig. 3, E and F), the average path length did not change significantly, but the clustering coefficient still underwent a biphasic change with increasing degree of sprouting. Therefore we could use sprouting-only networks (showing unchanged L but biphasic changes in C) to test whether biphasic changes in the clustering coefficient alone would conserve the biphasic changes in network hyperexcitability. Accordingly, we constructed functional model networks where mossy fiber sprouting occurred in the absence of hilar neuron loss. Again, before the actual simulations were performed, we first verified that the graphs of these sprouting-only functional model networks indeed showed the expected biphasic changes in absolute and relative clustering coefficient (green lines in Fig. 3, B and D), without significant changes in absolute and relative average path length (green lines in Fig. 3, A and C). (Note that these calculations were crucial because the functional model network contained 20 times fewer neurons than the number of nodes in the structural model network. Therefore topological changes taking place in the structural model cannot be automatically assumed to occur in the graph of the functional model.)

In response to perforant path stimulation, the sprouting-only functional model networks showed increasing granule cell activity with progressive sprouting, which spread to the entire model network in a self-sustained manner at 40–100% sprouting (Fig. 6). Detailed analysis of the activity in the sprouting-only networks (green lines in Fig. 5, A–D) showed that both the maximal duration of activity and the mean number of spikes fired per granule cell increased monotonically with increasing degree of sprouting and exceeded those in the corresponding sclerotic networks (compare black and green lines in Fig. 5, A and B; note the difference in scale of the y-axes), whereas the latency of the spread of activity to the most distant granule cells steadily decreased (green line in Fig. 5C). In contrast to the sclerotic networks (black lines in Fig. 5D), the local and long-range coherence of the granule cell firing remained similar to each other for all degrees of sprouting in the sprouting-only networks (green lines in Fig. 5D). The monotonic increase in granule cell firing and decrease in latency of activity propagation in the sprouting-only networks reflected the monotonic increase in excitatory drive to granule cells with progressive sprouting (68 nS increase per granule cell at maximal sprouting; green line in Fig. 5E), whereas inhibition remained constant (green line in Fig. 5F).

View larger version (35K): [in this window] [in a new window]   FIG. 6. Functional effects of sprouting without hilar neuron loss in the functional model network. A–F: raster plots of action potential discharges of granule cells in the functional model networks at increasing degrees of mossy fiber sprouting. Network activity was initiated as in Fig. 4. Note that maximal hyperactivity occurred at maximal sclerosis (for quantification, see Fig. 5, A–D).

Control simulations and tests of robustness

To examine the robustness of our results, we tested the effects of changes in structural and functional parameters on L and C values and on network activity in an extensive series of control simulations. These controls are described in appendixes B2 and B3 and illustrated in Figs. 7–9. The control simulations concerning structural parameters [APPENDIX B2, (1)–(5)] included tests of cell-type–specific changes in neuronal density along the septotemporal axis, inhomogeneity in connectivity along the transverse axis, axonal distributions at the septal and temporal poles of the model dentate gyrus, offset degrees of cell loss and sprouting, and the bilateral dentate gyrus model with both associational and commissural projections. The control simulations concerning functional parameters [APPENDIX B3, (1)–(3)] included increasing (doubling) the strength of inhibitory synaptic connections, inclusion of axonal conduction delays, and simulation of spontaneous instead of stimulation-evoked activity.

View larger version (41K): [in this window] [in a new window]   FIG. 7. Control simulations for structural parameters. A: cell-type–specific changes in neuronal densities along the septotemporal axis of the dentate gyrus, based on estimates from published data [see APPENDIX B2(1)]. Bin #1 is the septal end. B: 4 distinct boundary conditions, implemented to test the effects of skewing axonal distributions toward the septal and temporal poles of the model dentate gyrus [see APPENDIX B2(3)]. C: results of the calculations of L (C1) and C (C2), and of the simulations of activity (C3), for the various controls designed to test the effects of changes in structural parameters. Labels in C1 apply to all panels in C: "Base": base model used in the simulations described in RESULTS [corresponding to reduced septotemporal span, reduced total axonal length; see APPENDIX B2(3)]; "ST cell distributions": septotemporal cell distributions [illustrated in A and described in APPENDIX B2(1)]; "Transverse": inhomogeneous connectivity perpendicular to the septotemporal axis [APPENDIX B2(2)]; "Offset": offset (i.e., staggered) changes in the degrees of sprouting and hilar cell loss [APPENDIX B2(4)]; "Cons. ax. Length": conserved axonal length, with reduced septotemporal span; "Cons. ST span": conserved septotemporal span, reduced axonal length; "Cons. ax. length and ST span": both axonal length and septotemporal span conserved. Note that for the "offset" simulations, 70% sprouting with 90% hilar cell loss is plotted as 80% sclerosis for simplicity; similarly, 80% sprouting with 100% hilar cell loss is plotted as 100% sclerosis. Note that all 6 controls for structural parameters plotted in C yielded similar L and C values as those of the base model, and they all showed the characteristic decrease in activity with the progression of sclerosis from 80 to 100%.

View larger version (61K): [in this window] [in a new window]   FIG. 9. Control simulations for functional parameters. A: granule cell activity in functional model networks with double inhibition [maximal conductance of all inhibitory synapses indicated in Table 2 was doubled compared with the base network used in the rest of the simulations; see APPENDIX B3(1)]. B: granule cell activity with axonal conduction delays included in the network [an axonal conduction velocity of 0.25 m/s (Bartos et al. 2002) was implemented in the network; see APPENDIX B3(2)]. C: granule cell activity in functional model networks with spontaneous activity [simulated 10-Hz Poisson-distributed perforant path inputs were applied independently to all granule cells, all basket cells and 100 mossy cells, as a function of increasing amount of sclerosis; see APPENDIX B3(3)]. A1, B1, C1: 0% sclerosis; A2, B2, C2: 80% sclerosis; A3, B3, C3: 100% sclerosis. D: summary plot of granule cell activity for the base model used in the rest of the paper and for the 3 control simulations depicted in this figure. Note that, in spite of large changes in functional parameters, the characteristic biphasic shape of the changes in activity with sclerosis (the drop in activity with the progression of sclerosis from 80 to 100%) was present in all simulations.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX A1: ESTIMATION OF... APPENDIX A2: ESTIMATION OF... APPENDIX A3: FITS TO... APPENDIX B1: CONTROLS FOR... APPENDIX B2: CONTROLS FOR... APPENDIX B3: CONTROLS FOR... GRANTS ACKNOWLEDGMENTS REFERENCES

 
The following are major findings of this study. 1) The control dentate gyrus is a locally and globally well connected small-world network. 2) Structural alterations during epileptogenesis result in biphasic changes of small-world topology. Initially, and during the majority of the sclerotic process, the features defining the small-world characteristics of the dentate gyrus are enhanced (the dentate gyrus becomes "more small world" than in its control state). This is shown by a decreased relative L and increased relative C, indicating enhanced global and local connectivity. However, just before maximal sclerosis is reached, the relative L sharply increases as a result of the loss of the last long-distance projecting hilar cells, while relative C declines, leading to an overall decrease in the small-world characteristics. 3) Analyses of isolated excitatory and inhibitory graphs show that biphasic changes in small-world characteristics correspond to changing roles of topological factors: During submaximal sclerosis, the key factor is the predominantly local mossy fiber sprouting, which (as long as at least some long-distance projecting hilar cells survive) compensates for the loss of hilar cells (leading to a decrease in relative L, despite the loss of many long-distance connections). However, during the last stages of sclerosis, the loss of hilar neurons becomes the major topological determinant: as the last long-distance connections are removed, the network evolves into a more regular, lattice-like structure. 4) Simulations in the functional network show that changes in network topology predict the development of hyperexcitability: during submaximal sclerosis, when small-world characteristics increase, network hyperexcitability increases; however, during the last stages of sclerosis, when small-world characteristics decrease, the network hyperexcitability declines. 5) Sprouting-only networks, showing insignificant changes in L but biphasic changes in C, exhibited monotonic increases in hyperexcitability. These latter results further underline the importance of network topology by indicating that, in networks with concurrent sprouting and hilar cell loss, the biphasic changes in relative L (and the corresponding transition from a small-world to more regular network topology) are primarily responsible for the biphasic change in network hyperexcitability. Consequently, these results suggest that the survival of even a few hilar cells is critically important in maintaining networkwide hyperexcitability in the epileptic dentate gyrus.

Limitations of the models and robustness of the conclusions

Although our structural model was full scale and incorporated eight neuronal classes with cell-type–specific and topographic connections, the "virtual dentate gyrus" did not replicate the complete connectivity of the biological network. In most cases, specific components were not represented in the structural model because no precise data were available. For example, recent results indicate that local connection probability may be modified by intraclass correlations (Yoshimura and Callaway 2005; Yoshimura et al. 2005