Structural inhomogeneities in synaptic efficacies have a strong impact on population response dynamics of cortical networks and are believed to play an important role in their functioning. However, little is known about how such inhomogeneities could evolve by means of synaptic plasticity. Here we present an adaptive model of a balanced neuronal network that combines two different types of plasticity, STDP and synaptic scaling. The plasticity rules yield both long-tailed distributions of synaptic weights and firing rates. Simultaneously, a highly connected subnetwork of driver neurons with strong synapses emerges. Coincident spiking activity of several driver cells can evoke population bursts and driver cells have similar dynamical properties as leader neurons found experimentally. Our model allows us to observe the delicate interplay between structural and dynamical properties of the emergent inhomogeneities. It is simple, robust to parameter changes and able to explain a multitude of different experimental findings in one basic network.

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Dendrites form predominantly binary trees that are exquisitely embedded in the networks of the brain. While neuronal computation is known to depend on the morphology of dendrites, their underlying topological blueprint remains unknown. Here, we used a centripetal branch ordering scheme originally developed to describe river networks—the Horton-Strahler order (SO)–to examine hierarchical relationships of branching statistics in reconstructed and model dendritic trees. We report on a number of universal topological relationships with SO that are true for all binary trees and distinguish those from SO-sorted metric measures that appear to be cell type-specific. The latter are therefore potential new candidates for categorising dendritic tree structures. Interestingly, we find a faithful correlation of branch diameters with centripetal branch orders, indicating a possible functional importance of SO for dendritic morphology and growth. Also, simulated local voltage responses to synaptic inputs are strongly correlated with SO. In summary, our study identifies important SO-dependent measures in dendritic morphology that are relevant for neural function while at the same time it describes other relationships that are universal for all dendrites.

We investigate polyhedral 2k-manifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex k-Hamiltonian if it contains the full k-skeleton of the polytope. Since the case of the cube is well known and since the case of a simplex was also previously studied (these are so-called superneighborly triangulations), we focus on the case of the cross polytope and the sporadic regular 4-polytopes. By our results the existence of 1-Hamiltonian surfaces is now decided for all regular polytopes. Furthermore we investigate 2-Hamiltonian 4-manifolds in the d-dimensional cross polytope. These are the "regular cases" satisfying equality in Sparla's inequality. In particular, we present a new example with 16 vertices which is highly symmetric with an automorphism group of order 128. Topologically it is homeomorphic to a connected sum of seven copies of S 2 × S 2 . By this example all regular cases of n vertices with n < 20 or, equivalently, all cases of regular d-polytopes with d ≤ 9 are now decided.