Modelling the Human Brain as a directed graph A mathematician on the MathOverflow, Hans-Peter Stricker, recently(2016) considered the possibility that human brains may be modelled as directed graphs with neurons as nodes and synapses as edges. I think this makes sense as neurons are mostly unidirectional. 

Now, what I am curious about is whether there are ever errors in the developmental process such that there are 'collisions' between two neurons. I can imagine this occurring in two ways:


Is it possible for dendrites to connect to other dendrites?   
Alternatively, can axon terminals connect to other axon terminals?


If so, are there publications that estimate the natural frequency of such errors in a human brain? 

References:


Edward T. Bullmore and Danielle S. Bassett. Brain Graphs: Graphical Models of the Human Brain Connectome. 2011. 
Danielle S. Bassett and Edward T. Bullmore. Small-World Brain Networks Revisited. 2017.
Axo-axonic synapses and dendrodendritic synapses exist, and are not errors. They are particularly common in invertebrate nervous systems but are also found between particular cells in mammalian brains. I would say they are relatively understudied but not ignored.

True errors of the type you describe seem extremely unlikely to me. Synapses do not simply impinge on other fibers, there is extensive pre- and post-synaptic machinery, and bidirectional communication is necessary for proper synaptogenesis. Synapses are held together by connective proteins that are anchored in each membrane.

However, in terms of thinking of the brain as a "directed graph" one need not consider these uncommon synapses to have bidirectional connectivity: it is entirely possible, if not common, for cells to make traditional axon-dendrite synapses on another neuron that makes a reciprocal axon-dendrite synapse.

Graph theoretical approaches to understanding the brain are quite common, however, and certainly did not arise on StackExchange in 2016, as your own references make clear. There is no need for a directed graph to only have connections in a single direction, either. Any connectivity matrix can be interpreted in terms of graph theory.



Haydon, P. G., &amp; Drapeau, P. (1995). From contact to connection: early events during synaptogenesis. Trends in neurosciences, 18(4), 196-201.

Kawaguchi, Y., &amp; Kubota, Y. (1997). GABAergic cell subtypes and their synaptic connections in rat frontal cortex. Cerebral cortex (New York, NY: 1991), 7(6), 476-486.

Pinault, D., Smith, Y., &amp; Deschênes, M. (1997). Dendrodendritic and axoaxonic synapses in the thalamic reticular nucleus of the adult rat. Journal of Neuroscience, 17(9), 3215-3233.

Somogyi, P. (1977). A specific ‘axo-axonal’interneuron in the visual cortex of the rat. Brain Res, 136(2), 345-350.