X-Message-Number: 26033 From: 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 single 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 signal 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 a 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 with 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 signal 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 given 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 shift in the differential equation solution used. 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=26033