X-Message-Number: 25971
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 
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 
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 
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 
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 
(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.

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