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I assume that human brains can be considered as directed graphs with neurons as nodes and synapses as edges. I explicitly don't want to consider the weights, the dynamics of neural activity (based on the weights), and the adjustment of weights (learning) - just brains as static unweighted finite directed graphs.

Sensor neurons may be those having in-degree 0, actor neurons may be those having out-degree 0. (0 meaning "essentially 0".)

Considering human brains as finite directed graphs, for each question concerning finite directed graphs there should be an answer with respect to human brains.

Such questions might be:

  • How long is the shortest path from a sensor to an actor neuron?

  • How long is the longest (direct) path from a sensor to an actor neuron?

  • What is the (global/local) layer structure (on different levels of granularity)?

  • What is the (global/local) cycle structure (on different levels of granularity)?

I find it hard to get answers to such questions considering human brains as directed graphs, because neuro-scientists don't think in terms of graphs, but for example in terms of signal paths and neuro-anatomy. But then - for them - "anything goes", and "everything is connected to everything" - which is not very helpful.

I would be very glad for any reference treating (formally) human brains as directed graphs.

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  • $\begingroup$ Do these questions have any physical interpretation? As in they describe some behavioural phenomena, or neuroscientific event? This sounds really neat, I'm just wondering if you have some scientific reasoning for asking this question. $\endgroup$ – user78249 Nov 29 '16 at 1:01
  • $\begingroup$ The questions stand for themselves: I really want to know what is the longest path from any sensory to an actor neuron. And I really just want to learn more about the layer and cycle structure(s) of the human brain. $\endgroup$ – Hans-Peter Stricker Nov 29 '16 at 1:05
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    $\begingroup$ A 2011 survey calls these directed brain graphs. They say little work has been done specifically on directed brain graphs, but that this "will be a priority for future technical innovation." Bullmore, Edward T., and Danielle S. Bassett. "Brain graphs: graphical models of the human brain connectome." Annual Review of Clinical Psychology 7 (2011): 113-140. $\endgroup$ – Joseph O'Rourke Nov 29 '16 at 1:59
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    $\begingroup$ Of course, the "human brain" is only a special kind of finite directed graph of which it would be interesting to know its characteristic features. $\endgroup$ – Hans-Peter Stricker Nov 29 '16 at 8:38
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    $\begingroup$ Fruit fly $\endgroup$ – AHusain Nov 29 '16 at 8:57
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There is empirical evidence that the connectivity in the brain has the characteristics of a directed small-world network.

Small-world directed networks in the human brain: Multivariate Granger causality analysis of resting-state fMRI (Wei Lao et al., 2010):

Small-world organization is known to be a robust and consistent network architecture, and is a hallmark of the structurally and functionally connected human brain. However, it remains unknown if the same organization is present in directed influence brain networks whose connectivity is inferred by the transfer of information from one node to another. Here, we aimed to reveal the network architecture of the directed influence brain network using multivariate Granger causality analysis and graph theory on resting-state fMRI recordings. We found that some regions acted as pivotal hubs, either being influenced by or influencing other regions, and thus could be considered as information convergence regions. In addition, we observed that an exponentially truncated power law fits the topological distribution for the degree of total incoming and outgoing connectivity. Furthermore, we also found that this directed network has a modular structure. More importantly, according to our data, we suggest that the human brain directed influence network could have a prominent small-world topological property.

More recent studies of the human brain as a directed graph are summarised in section 7.3 of this review article.

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Human brains as graphs were considered by Kolmogorov and his students. Some results were published in the article О реализации сетей в трехмерном пространстве А.Н. Колмогоров, Я.М. Барздинь - Проблемы кибернетики, 1967. You can find English translation of this work in the book A. N. Kolmogorov, Selected works - Information theory and the theory of algorithms, page 194.

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