```X-Message-Number: 26032
From:  var s1 = "Azt28"; var s2 = "aol.com"; var s3 = s1 + "@" + s2; document.write("<a href='mailto:" + s3 + "'>" + s3 + "</a>");
Date: Sat, 16 Apr 2005 08:08:16 EDT

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

Boolean, Differential and Stochastic. Here I look at the first differential
model
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
brain
without functional reduction. It is possible to play with the cable theory
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
defined
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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