PLoS supp. material by Hermann Cuntz
Combining Kirchhoff’s junction law for electrical circuits with simple linear algebra,
a matrix can be derived which describes the equivalent circuit of a tree.
If we cut the circuit at node 3 in the above example the following matrix appears:
Dividing a vector or matrix of input currents I by
the conductance matrix M results in potential vectors V
(or matrix respectively) according to Ohm’s law.
Simply taking the inverse of the conductance matrix M results
in the steady state electrotonic signature of the tree:
VSSE = M-1
This electrotonic signature (see “sse_tree”)
describes well the electrotonic compartment-alization of a neuronal tree.
In this case the matrix of input currents I is simply the identity matrix.
Currents of 1 (nA) are therefore injected one at a time in each node
in subsequent columns or rows.
The symmetrical square matrix VSSE
contains in each column or row the potential distribution
in all nodes following the current injection in the corresponding node
(i.e. the current transfer).
The diagonal therefore contains the local input resistances
since there the potential change is measured in each node
resulting from current injection into the same node.
Red squares correspond to sub-trees with increased
The electrotonic signature therefore follows closely on the adjacency matrix
(as can be seen from the relationship between M and dA).
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