X-Message-Number: 25971 From: Date: Mon, 4 Apr 2005 03:02:18 EDT Subject: Uploading technology (2.iii.0). Uploading technology (2.iii.0). Here I continue to define the mathematical basis of neuron modeling for reference purpose. In i/ the main conclusion was to use the Hindmarsh-Rose model for dendrite tree modeling. In ii/ the most nonlinear part in dendrite spine head and cell body was ascribed to van Kampen's omega Taylor serie, reduced in a first step to Langevin's equation (This holds for gene regulation but let unanswered the problem of signal transmission in the cell body). In dendrite head the volume is very limited, so there is no need to take into account the diffusion and gradient of molecular distribution. Yet, Ca++ ions may be an exception here. The tools of choice for that case is particular, I'll come back on that subject another time. Spine neck seems to have as main function to suppress any diffusion from the spine head to the dendrite tree, so here is no need for diffusion here neither. The cell body on the other hand is quite another problem. To bypass diffusion here would be a big mistake. In cells, diffusion is in fact "quasi diffusion" in most case. The true diffusion is the simple effect of statistical mechanics producing a random walk of molecules in the solvent, mostly water. This is the so-called brownian motion explained one century ago by Einstein. At macroscopic scale this produce the Fick's law (1). This simple process fits only nonactive biological molecule, that is, the one without interest or nearly so. When dealing with ions such Ca++, diffusion is largely dominated by interaction with buffering molecules and the true speed of the process is far slower than "free" diffusion. For big molecules, in small number, diffusion is largely influenced by stochastic processes. The basic tools of diffusion modeling is then a set of differential equation broken into a sum of Dirac's delta distribution. To take into account the random fluctuation in molecule number, the tool is the "Monte Carlo Green's Function". The Monte Carlo method is a statistical analysis of a number of discrete cases drawn at random or quasi random, because computers are deterministics. The Green's fuction may be tought of as the inverse process of a differential equation, itself a backward recipe: If you have some data and a rule to combine them, you compute a function. When you have the data and the result and may be how it behave, you must find the function, that is a differential equation. Now if you break a differential equation into generalized functions or so called distributions, the inverse process gives a Green's function, it is an inverse distribution. This is not the end of the road: The body cell is heavily compartmentalized and the method must be used independently on each compartment in two or three dimensions. As a general rule, there is 3 dimensions for each compartment, this bring rapidly a space with very many dimensions and a corresponding explosion in the computing power request. Because even simple problems here are very computer hungry, this is particularly unwelcome. The solution may be to reduce the number of dimensions with the methods of variational dynamics (2). This produce very complex equations with strange behaviors, but it reduces the number of dimensions. Variational analysis is outmoded now, it was useful before the advent of computer. Now, nobody would spend many weeks or month to reduce the computing load of a problem. The general answer is: Buy a more powerful computer and run head on in the number mountain. Here, we hit the limit of that process, even computer are overflown by the problem and reducing the computing load is a good idea. Some package such Mathematica can be used to reduce the analytic task. This looks very complex and it is indeed. On the good side, such processes are somewhat on the sideway of the neuron information processing and it is not a prime necessity to compute all that independently for each neuron at each microsecond. A set of generic solutions can be precomputed and storred in a memory. What will be done on the user side is simply to take a value in the memory or interpolate between two values. To summarize, the cell body brings a new stage of complexity, (quasi)diffusion can't be neglected here, the corresponding differential equations are broken into a set of Dirac's delta distributions. When dealing with big, rare molecules, the stochastic aspect of diffusion must be simulated with Monte Carlo Green's functions. The higly structured space of the cell soma will ask for very many dimensions, these will be reduced by an analytic study using variational dynamics. (1) Fick A. (1885) Ueber diffusion. Ann. Phys. Chem. vol. 94; p. 59 - 86. (2) see for example : The Variational Principle of Mechanics, Cornelius Lanczos, Dover 1986. Yvan Bozzonetti. Content-Type: text/html; charset="US-ASCII" [ AUTOMATICALLY SKIPPING HTML ENCODING! ] Rate This Message: http://www.cryonet.org/cgi-bin/rate.cgi?msg=25971