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Most current studies of neuronal activity dynamics in cortex are based on network models with completely random wiring. Such models are chosen for mathematical convenience, rather than biological grounds, and additionally reflect the notorious lack of knowledge about the neuroanatomical microstructure. Here, we describe some families of new, more realistic network models and explore some of their properties. Specifically, we consider spatially embedded networks and impose specific distance-dependent connectivity profiles. Each of these network models can cover the range from purely local to completely random connectivity, controlled by a single parameter. Stochastic graph theory is then used to describe and analyze the structure and the topology of these networks.

The architecture of any network can be an essential determinant of its respective function. Signal processing in the brain, for example, relies on a large number of mutually connected neurons that establish a complex network [

Networks with a probabilistically defined structure represent, from a modeler’s perspective, a viable method to deal with this lack of detailed knowledge concerning cell-to-cell connections [

Random graphs [

Left: reconstruction of a pyramidal cell stained in a tangential slice of the rat neocortex (top view). Middle: schematic 2D section representing a spatially embedded network composed of locally (red lines) connected pyramidal cells (black triangles). Right: different types of abstract networks.

The graph-theoretic analysis of cortical networks raises the following problem: graphs usually do not deal with space (right part Figure

Altogether, we face a spatially embedded and very sparsely connected network, where only a very small fraction of neuron pairs are synaptically coupled to each other directly. What is the impact of these general structural features of synaptic wiring in the cortex? Do these features matter in determining the global topology of the network? Sparse couplings save cable material, but they also constrain communication in the network. Can the sparsity, in principle, be overcome by smart circuit design? Likewise, admitting only neighborhood couplings saves cable length but increases the topological distance between nodes in the network, that is, the number of synapses engaged in transmitting a signal between remote neurons becomes quite large [

Preliminary results of this study have been presented previously in abstract form [

We considered network models that comprised neurons with directed synaptic connections. Therefore, our cortical networks were represented by directed graphs

List of randomness parameters used to construct the spatially embedded RPNs.

SW: | 0 | 0.01 | 0.02 | 0.05 | 0.1 | 0.2 | 0.5 | 0.8 | 1 |

FN: | 1 | 0.99 | 0.98 | 0.95 | 0.9 | 0.8 | 0.5 | 0.2 | 0.01 |

FN: | 0.0611 | 0.0614 | 0.0617 | 0.0627 | 0.0644 | 0.0683 | 0.0864 | 0.1366 | 0.5 |

12 | — | 11.76 | 11.4 | 10.8 | 10.2 | 12.00 | 12.00 | 11.9 | |

GN: | 1 | — | — | 0.95 | 0.9 | 0.8 | 0.5 | 0.2 | 0.05 |

GN: | 0.0432 | — | — | 0.0443 | 0.0455 | 0.0483 | 0.061 | 0.0965 | 0.197 |

Left: simple ring graph composed of 6 nodes. Middle: the corresponding adjacency matrix. Right: scheme describing the construction of spatially embedded networks with distance-dependent connectivity. Each node (red dots) has a connectivity disk (filled circle); blue arrows indicate periodic boundary conditions (torus topology).

Each neuron was situated in a quadratic domain of extent

In a network with no long-range connections, nodes placed within a circular neighborhood of radius

We considered the following three families of networks, each spanning the full range from regular to random connectivity.

Fuzzy neighborhood (FN) network: this model assumed uniform connectivity of probability

Small-world- (SW-) like network: again starting from

Gaussian neighborhood (GN) network: Gaussian profiles were used to define a smooth distance-dependent connection probability, adjusted to the connectivity parameters of the FN networks. The corresponding parameter pairs were

The following descriptors were used to characterize and compare the network models described above. Most quantities are well established in the context of graph theory (see, eg., [

(a) Degree distributions and correlations: counting incoming and outgoing links for each node of a graph yield an estimate of the distribution of in-degrees

(b) Small-world characteristics: the cluster coefficient describes the probability that two nodes, both connected to a common third node, are also directly linked to each other. Let

(c) Wiring length: since we deal with spatially embedded networks, any pair of nodes

(d) Eigenvalues and eigenvectors: for any graph

Localization of four sample eigenvectors of spatially embedded graphs. The squared value of each component of an eigenvector is represented by a rectangle of proportional area, centered at the position of the corresponding node. Top left: FN random position network with

We employed several characteristic network properties to compare different types of spatially embedded networks (FN, SW, and GN). Comparing FN and SW connectivity, we aimed to analyze the effect of unconstrained long-range connections, as opposed to the compact FN connectivity. We also asked if GN connectivity provides an appropriate compromise, involving long-range links combined with a compact local connectivity range. We focused on networks with random node positions (RPN), while the results for lattice position networks (LPNs) and the corresponding 1D ring graphs are only discussed in case of a significant deviation.

In the FN, SW, and GN scenarios, networks with random node positions exhibited binomial distributions for both the in- and out-degree (Figure

Top left: binomial out-degree distributions (gray) for RPNs based on FN and SW connectivity for different parameter settings. The fitted binomial distribution is superimposed (black). Top right: degree correlations depending on the randomness parameters

Figure

The three histograms in Figure

In this section, most results are shown for RPNs. Concerning the average shortest path length and the mean distance of connected nodes, any differences between RPNs and LPNs were negligible. Only the cluster coefficient was significantly higher in case of RPNs; for a detailed analysis of this issue, see [

The well-known characteristic feature of small-world networks is the

Top left: cluster coefficient

Figure

We also computed the degree-dependent cluster coefficient

Degree-dependent cluster coefficient for RPNs (left) and LPNs (right) with SW (red symbols) and FN (blue symbols) connectivity. Each figure shows the results for several different values of the parameters

To summarize, FN and GN models did not exhibit any small-world characteristics. There was no reduction of

Concerning the eigenvalue distribution of the adjacency matrix, we again found characteristic differences due to the spatial embedding, especially in the case of near-regular connectivity. We observed, however, again only small deviations between different types of connectivity.

Figure

Eigenvalue density of FN (left) and SW (right) RPNs ranging from local (top) to random (bottom) networks. Top: real eigenvalue spectrum of a (symmetric) locally connected network (

Complex eigenvalue density for LPNs with FN (

For both the FN and SW scenarios, the distribution of eigenvalues smoothly changed its shape from circular (most eigenvalues complex) with radius

Although the spectra of FN and SW networks were quite similar, the average spatial concentration of eigenvectors turned out to be a quite sensitive indicator for both the type of spatial embedding (RPN versus LPN) and the type of wiring (FN versus SW) assumed for the construction of the graph. Figure

Locality of eigenvectors for RPNs and LPNs, either with FN (left) or SW (right) connectivity. Top: entropy

In general, the eigenvectors corresponding to the largest absolute values of

Figure

We introduced two families of network models, each describing a sheet, or layer, of cortical tissue with different types of horizontal connections. We assumed no particular structure in the vertical dimension. Neurons were situated in space, and the probability for a synaptic coupling between any two cells depended only on their distance. Both models could be made compatible with basic neuroanatomy by adjusting the parameters of the coupling appropriately. Both families of networks spanned the full range from purely local, or regular, to completely random connectivity by variation of a single parameter. The paths they took through the high-dimensional manifold of possible networks, however, were very different.

The first model (fuzzy neighborhood) assumed a homogeneous coupling probability for neurons within a disk of a given diameter centered at the source neuron. The probability was matched to the size of the disk such that the total connectivity assumed a prescribed value. The related Gaussian neighborhood model was based on similar assumptions but its smoothly decreasing connection probabilities were defined by Gaussian profiles, adapted to those of the fuzzy neighborhood model. For very small disks, only close neighbors formed synapses with each other, and, for very large disks spanning the whole network, couplings were effectively random. The second model (small world) started with the same narrow neighborhoods but departed in a different direction by replacing more and more local connections with nonlocal ones, randomly selecting targets that were located anywhere in the network.

For most models considered in this study, the initial random positioning of neurons in space guaranteed that both in-degrees and out-degrees had always the same binomial distribution, irrespective of the size of the disk defining the neighborhood and irrespective of the number of non-local connections. This means that none of the statistical differences between the various candidate models described in the paper can be due to specific degree distributions. This is in marked contrast to the original demonstration of the small-world effect in ring graphs [

The first main result of this study is that networks residing in two dimensions—very much like one-dimensional ring graphs [

It seems reasonable to assume that, in neocortex, the length of a cable realizing a connection is roughly proportional to the physical distance it has to bridge. The second main result of this study is that the length of the average shortest graph-theoretical path was always inversely related to the total length of cable that is necessary to realize it (Figure

What conclusions can be drawn from graph spectral analyses? First of all, the complex eigenvalue spectrum of the adjacency matrix of a graph is a true graph invariant in the sense that any equivalent graph (obtained by renaming the nodes) has exactly the same spectrum. To some degree, the opposite is also the case: significantly different graphs give rise to differently shaped eigenvalue spectra. Empirically, it seems that similar graphs also yield similar spectra, but a rigorous mathematical foundation of such a result would be very difficult to establish. So we informally state the result that the shape of eigenvalue spectra reflects characteristic properties of graph ensembles, very much like a fingerprint. With an appropriate catalog at hand, major characteristics of a network might be recognized from its eigenvalue spectrum.

More can be said once the eigenvalue spectrum is interpreted in an appropriate dynamical context. Linearizing the firing rate dynamics about a stationary state allows the direct interpretation of eigenvalues in terms of the transient dynamic properties of an eigenstate. Real parts give the damping time constant, and imaginary parts yield the oscillation frequency. Although some care must be taken to correctly account for inhibition in the network [

Finally, we would like to stress once more the importance of identifying characteristic parameters in stochastic graphs and their potential yield for the analysis of neuroanatomical data. Measurable quantities, or combinations of such characteristic numbers, could be of invaluable help to find signatures and to eventually identify the type of neuronal network represented by neocortex.

The authors thank A. Schüz and V. Braitenberg for stimulating discussions. This work was funded by a grant to N. Voges from the IGPP Freiburg. Further support was received from the German Federal Ministry for Education and Research (BMBF; Grant no. 01GQ0420 to BCCN Freiburg) and the 6th RFP of the EU (Grant no. 15879-FACETS).