Topographical maps as complex networks

The neuronal networks in the mammalian cortex are characterized by the coexistence of hierarchy, modularity, short and long range interactions, spatial correlations, and topographical connections. Particularly interesting, the latter type of organization implies special demands on developing systems in order to achieve precise maps preserving spatial adjacencies, even at the expense of isometry. Although the object of intensive biological research, the elucidation of the main anatomic-functional purposes of the ubiquitous topographical connections in the mammalian brain remains an elusive issue. The present work reports on how recent results from complex network formalism can be used to quantify and model the effect of topographical connections between neuronal cells over the connectivity of the network. While the topographical mapping between two cortical modules is achieved by connecting nearest cells from each module, four kinds of network models are adopted for implementing intramodular connections, including random, preferential-attachment, short-range, and long-range networks. It is shown that, though spatially uniform and simple, topographical connections between modules can lead to major changes in the network properties in some specific cases, depending on intramodular connections schemes, fostering more effective intercommunication between the involved neuronal cells and modules. The possible implications of such effects on cortical operation are discussed.

Figure 1

Diagram representing the four typical network modules used in this work: preferential attachment (a), long range (b), random (c), and short range (d). The spatial distribution of nodes is the same for all diagrams. We considered $L=50$, $p=0.1$, $\gamma =0.01$ (i.e., $N=25$ nodes), and $T=0.03$ for the cases (b) and (d).

Figure 2

Diagram representing unidirectional topographical connections (dashed lines) from nodes of module B to nodes of module A, which are separated by distance $a$.

Figure 3

Diagram representing two network modules (A and B) separated by distance $a$ and connected bidirectionally by topographical connections (dashed lines).

Figure 4

The average and standard deviation of the shortest path lengths for unidirectional IMC networks with $p=0.1$. We consider four different modular architectures: preferential attachment (a), long range (b), random (c), and short range (d). Filled squares correspond to topographic IMC and open circles to random IMC. The scale of the vertical axes in (b) and (d) panels are the same as in (a) and (c), respectively.

Figure 5

The average and standard deviation of shortest path lengths for unidirectional IMC networks with $p=0.3$. We consider four different modular architectures: preferential attachment (a), long range (b), random (c), and short range (d). Filled squares correspond to topographic IMC and open circle to random IMC. The scale of the vertical axes of (b) and (d) panels are the same as that of (a) and (c), respectively.

Figure 6

The average and standard deviation of shortest path lengths for bidirectional IMC networks with $p=0.1$. We consider four different modular architectures: preferential attachment (a), long range (b), random (c), and short range (d). Filled squares correspond to topographic IMC and open circles to random IMC. The scale of the vertical axes of (b) and (d) panels are the same as that of (a) and (c), respectively.

Figure 7

The average and standard deviation of the shortest path lengths for bidirectional IMC networks with $p=0.3$. We consider four different modular architectures: preferential attachment (a), long range (b), random (c), and short range (d). Filled squares correspond to topographic IMC and open circles to random IMC. The scale of the vertical axes in (b) and (d) panels are the same as in (a) and (c), respectively.

Figure 8

Fraction of pairs of nodes without a path connecting them, $⟨\pi ⟩$, versus the standard deviation of the shortest path lengths, $\Delta \delta$, for unidirectional (a) and bidirectional (b) connection types. We consider PAT networks with topographical IMC (filled squares) and random IMC (open circles); and LNR networks with topographical IMC (asterisk symbols) and random IMC (open rhomboids).