Talk:Scale-free neocortical dynamics
Dear Dr. Freeman,
There are a few points I would request that you clarify for the final paper.
You mention that Bollobas and Riodan derived a measure of diameter of the scale-free network of nu nodes. Their equation is log(nu)/log(log(nu)). You calculate nu as the number of synapses in the network, rather than the number of cells and derive a number of 5e13. My calculations of ln(5e13)/ln(ln(5e13) results as 9 for natural logarithms and 12 for log base 10, both are different from your result of 11. I am also confused; Generally, nodes are referred to as the number of cells, not synapses. Is nu a node or a connection? I think the diameter should reflect the size of footprint of connections from a neuron, this equation does not take into account the distribution or even the number of connections per neuron. Therefore, I’m not really sure what it is a measure of. Furthermore, I do not follow how you come to the conclusion that a diameter of 11 justifies a 3-leevl hierarchy of nodes. Could you please elaborate on this point so that the conclusion is self contained within the article and does not require reading Bollobas and Riordan to understand the point.
I like the graph and section about the different methods of connecting networks. You mention that the small world network has a form of predominantly local connections and a few long distance connections. This network is a small world network (SWN) as defined by Watts and Strogatz, but it is not the only SWN that they defined. Their definition of SWN is based on the measure of a small connection distance between any pair of neurons and a large clustering coefficient regardless of the connectivity in the network. By their definition, scale-free networks are also small world netwoks. I’m not sure how to best clarify this point, but I want to make sure that people don’t think SWNs are limited to this, or defined by this type of connectivity.
I don’t like the use of the sentence, “The simplest way to model the intervening central steps…”. I am always dubious of someone who tells me what is “simplest”.
“Like a brain, a random graph is not static.” When I think of a random graph, I think of a network that has been wired together randomly. Classically, I don’t think of a random graph as a dynamic network. It becomes clearer in the next sentence that you are comparing the construction of a dynamic graph with the development to of the brain. I think this needs to be clarified in the lead sentence.
I would be cautious about the conclusion “…, it is safe to conclude that the distribution of cortical structural connection lengths are power-law.” First, I think you should say that the self similarity of dendritic trees is across scale, to clarify how this supports the power-law connectivity. Secondly, since this conclusion is “pending further investigation” I would not say it is a “safe” conclusion, but perhaps a “parsimonious” one.
In “1/f^a PSD with a near 2”, the “a” should be italicized to indicate that is a variable and not a letter. You write, “Preferntiality in scale-free network, …”, Do you mean “Scale-free networks preferentially make hubs…” by the way they are connected?
Your section on and mention of state transitions throughout the text are not really a part of scale-free networks. It is a feature that is enabled by scale-free networks and I think makes an interesting hypothesis of how the brain works and why this kind of networking is important, but should probably be left to another article on State Transitions in Neuronal Dynamics and linked to it through this article and pair this article down to the essentials of a scale-free network.
Your final section on Focal lesions needs more support. According to scale-free hypothesis, each level of the network should have hubs of the size of the network. I am not sure, taking the example of pyramidal cells in the hippocampus, that some cells have order of magnitude more synapses than the others. Would you argue that the hub cells are probably of a different population or that there is scale free connectivity of all populations and pyramidal cells have great variability in numbers of synapses? You imply that the brain stem must be a hub system, I think you need to support this hypothesis with anatomical evidence that theses regions project to more regions than others, or to more cells and that it is not a difference in type of projection than quantity. My retort might be that a computer has an off switch which could stop all activity, does it make it a hub? I don’t think it does, the CPU probably is not scale free, and the fact that the system can go through a state transition by turning off the power doesn’t mean that it has a hub system. If you are to claim that the brain stem is a hub, it requires more evidence or references.
Reply to Dr. Netoff
Dear Dr, Netoff, I thank you for your many suggestions for corrections and clarifications, most of which I have adopted in my current draft. I demur on two points. First, your suggestion that I de-emphasize state transitions is counter to my intent. This is not an entry on "Scale-free Networks"; it is an entry describing the difficulties and importance of applying concepts from random graph theory to neocortical dynamics. The state transition is of paramount importance; the scale-free substrate connectivity is of major value for explaining the way in which extremely large domains of cortex (large in comparison to the size of most neurons) undergo sudden phase re-settings and re-synchronizations independently of distance and size. Second, your suggestion that I elaborate on evidence for hubs in the sometimes catastrophic effects of focal lesions is well advised, except that this entry is already very long and complex. I think it is important to state the availability of neurological evidence, but to leave the details to another entry of the encyclopedia. Certainly the anatomical evidence is widely available, and certainly I do not equate the classic but now defunct "reticular activating system" to an "on-off" switch in a computer, though the I/O divergence-convergence of its multiple coponents is legendary. With appreciation, W J Freeman