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R2
10 seconds
R3
10 seconds
R15
10 seconds
L10
10 seconds
Figure 2.6.
Firing patterns of identified neurons in Aplysai's ganglion are portrayed. R2 is normally silent,
R3 has a regular beating rhythm, R15 a regular bursting rhythm and L10 an irregular bursting
rhythm. L10 is a command cell that controls other cells in the network. Reproduced with
permission from [
36
] Fig. 4.
patterns result from differences in the types of ionic currents generated by the mem-
brane of the cell body of the neurons. The variety of dynamics observed in neurons as
depicted in Figure
2.6
has nearly as many scales as that in cell morphology.
It is probably worth speculating that the enhanced complexity in the firing pattern of
L10 in Figure
2.6
may be a consequence of its greater role within the neuron network
of which it is a part. The distribution in the time intervals between firings is also of
interest, particularly for those that fire intermittently such as L10. It does not seem rea-
sonable that the various kinds of complexity observed in these data, namely the irregular
branching in space and the multiplicity of scales in time, are independent of one another.
Interrelating these two kinds of complexity is one of the goals of network science. To
reach this goal, let us begin exploring some of the modeling associated with complexity
of branching. For this we go back a few hundred years to the ruminations of Leonardo
da Vinci on natural trees and scaling in his notebooks [
64
].
2.1
Natural trees and scaling
The simplest type of branching tree is one in which a single conduit enters a vertex and
two conduits emerge. The branching growth of such trees is called a dichotomous pro-
cess and its final form is clearly seen in botanical trees, neuronal dendrites, lungs and
arteries. In addition the patterns of physical networks such as lightning, river networks
