X-Message-Number: 26033
Date: Sat, 16 Apr 2005 08:57:17 EDT
Subject: Uploading (2.ii.1b) cable theory.

Uploading (2.ii.1b) cable theory.
There is the second and last part of the cable theory:
What is interesting here, is that such an equation can be solved by two 
different and unequal ways. The first and most classical one try to cut the 

equation vith two variables, X and T into two simpler equations, each with a 
variable. This is the so called variable separation method. It produces a 

familly of smooth solutions. If you run NEURON on your computer, you'll see the
result: An inpulse at a tip of the dendrite tree will generate a dull, flat 

signal at the soma level. This is not what we could wait for. How can such a 
can have any effect? Cooperation between branches  are assumed to produce a 
more stronger transmission, but that fall short of the requested amplitude and 
time definition. Conclusion: the cable model is a poor contender to give a 

good picture of the neuron working... It is a passive, dissipative component in 
poor conductor.
In fact, the problem is not with the model. It rests with the chosen equation 
solutions. A more general solution of a partial differential equation is 
produced by broking the equation in an infinite set of distributions or 

generalized functions. The simplest distribution is the delta function: a spike 
infinite height, zero width and unit surface! Taken as the elementary 

differential equation, its solution is a gaussian function, giving a bell 
shapped curve.
What that means for a cable element? The simplest solution will be a set of 
regularly spaced spikes, what is called a cha distribution. The name come from 
the russian letter cha, looking as a rake. To find the mathematical solution, 
it suffice to put a bell curve at the place of each spike. If the curves are 
near each other, they'll produce a sum nearly continuous. That is, the signal 
don't decay, it remains the same whatever the distance. It is regenerated at 
each bell curve.
The axon, the output organ of the neuron is very large in molusks. It must be 
so to have a minimum resistivity and allows a fast signal on a long distance. 
Molusks know only the variable separation differential solution, as do cable 
theory simulators on computers. Vertebrates species have myelinated axons with 
regularly spaced naked sections, the Ranvier nodes. Each node is a bell curve 
generator regenerating the signal. The vertebrate have invented the Green's 
function solution of their differential equation!
What about dendrites? There is no myelin, so a simple bundle of ionic 

channels can't be a solution. On the other hand, a dendrite spine could fire a 
when there is a passing membrane depolarization. It could works as a bell 

shape generator. This system has the possibility to suppress the signal damping
but it can amplify it too! The output of the dendrite section may be stronger 
than the input. Assume a typical unbranched dendrite element has room for 20 
spines. We could code it with a 20 bits "word". In a given place, "0" would 
means no spine and "1" there is a spine. A segment of that kind would have one 
million solutions, from twenty "0", the no spine case, the only one taken into 
account by present day simulators, up to 20 "1", the full spine ampifier.
We have there  an important element: Most dendrite spines are not linked to a 
terminal axon in a synapse, they are amplifier, or signal regenerators along 
the dendrite. That is the prediction of the Green's function solution of the 
partial differential equationof the cable theory. Indeed, electron microscope 
imaging of dendrite elements display a rough "cable" with clustered spines.
My feeling is that most of the neuron memory capacity is here: More than the 
signal generation at the synapse, it is the signal damping, processing or 

amplification along the dendrite sections that define the memory linked to a 
synapse link.
Most neural networks define a weight for the signal at the synapse but they 
don't take into account the memory potential of the dendrite tree itself. 

Implement that in an electronics device is not a problem, it simply needs a 
in the differential equation solution used.
Yvan Bozzonetti.

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