X-Message-Number: 26032 From: 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 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 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 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. Content-Type: text/html; charset="ISO-8859-1" [ AUTOMATICALLY SKIPPING HTML ENCODING! ] Rate This Message: http://www.cryonet.org/cgi-bin/rate.cgi?msg=26032