X-Message-Number: 26032
Date: Sat, 16 Apr 2005 08:08:16 EDT
Subject: Uploading (2.ii.1a) cable theory.

Uploading (2.ii.1a) cable theory.
In Uploading (2.ii.0) I have presented the 3 basic modeling domains : 

Boolean, Differential and Stochastic. Here I look at the first differential 
proposed, the so-called cable theory. The objective is to show that the 

understanding of a mathematical model can reduce the complexity of the uploaded 
without functional reduction. It is possible to play with the cable theory 
using the simulator "neuron". It can be downloaded free, see: 
http://www.neuron.yale.edu or: http://neuron.duke.edu .
The cable model assume a cylindrical conductor with some resistivity, a 

bounding tube with another resistivity, far larger than the internal one, and an
exterior medium with again low electrical resistivity. An electrical charge 
injected in the internal conductor can do three things:
1/ it can flow along the internal conductor until it dissipates into heat 
under the effect of resistivity.
2/ it can go througout the high resistance boundary membrane and flow into 
the exterior medium.
3/ it can be locked against the inner side of the membrane and bring an equal 
charge of the opposite sign on the outside. this is the capacitor effect.
Depending on the internal resistance (or its inverse, the conductance), the 
membrane r sistance and capacity, there will be a lenght L producing a current 
reduction equal to 1/e where e is the basis of the natural Log. e = 2, 

71828... In the same way, there is a given time called the time constant tc 
by the product of the resistance and the capacity of the boundary membrane. A 

signal shorter than tc will be transmitted, one longer will leak heavily in the
outside medium. To be sure, there is too a time constant given by the product 
of the internal resistance and the membrane capacity. This time constant, Tc, 
would block any transmission shorter than it. So a good cable has a fairly 
high membrane resistivity and low internal one.
If the cable under study has a lenght x, it is more natural to redefine it in 
units of L and write: X = x/L. In the same way, a time t will be recast in T 
= t/tc. The cable theory then says that a potential V produce a signal obeying 
the equation:
d^^2(V)/dX^^2 - V - dV/dT = 0; "d"  is in fact the "round d" of partial 

derivative. This is what is called a second order partial differential equation.

Don't bother to solve it, it is here only as an illustration. You can play with
"NEURON" as a monkey with a typewriter without knowing what you do. Well, you 
don't go far this way, that is why the next part will give a feeling of what 
can be extracted from the theory.
Yvan Bozzonetti.

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