Thursday, June 22, 2017

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Insights from the Blue Brain's latest paper on Cliques of Neurons bound into cavities ...

Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function

Front. Comput. Neurosci., 12 June 2017 |

I think the paper is a good food for thought, it directs for some advanced/interesting areas of mathematics such as Algebraic Topology.

Concepts and ideas to think about:
DAG/graphs, (directed) cliques, simplex/simplices, directed simplical complexes, sink/source (also vector calculus and gradients), cavities, dynamics of the above.

Betti number, n-th B.n., cavity formation

nb - n-th Betty number, ... number of n-dim cavities

Sink - last neuron, Source - first neuron. 

locally - simplices 
            cavities - globally 

directed flag complex
 shape - topology

Simplex ~ triangle in any dimension. Maximal simpl.Dim. ~ number of linearly-independent vectors?; not part of a higher dim. simpl.

Topology - connectivity, homology

formation-disintegration  -- rise of dimensionality, then reduction
random matrix
structural m.
transmission-response matrix
pre-, post- synaptic spike , period 10, 20 ms ...

cavity - homology class, fully enclosed by directed simpl. - dim.

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