# Speed of passive transport of ions in the dendrite

I tried hard to get them, but I didn't get any numbers. My (subjectively) best Google search was for "speed propagation post-synaptic potential".

My question is:

How fast are ions transported passively inside the dendritic tree (and soma), probably depending on their atomic mass, charge, and of the local geometry?

Especially I want to know when a post-synaptic potential generated x μm away from the axon hillock "arrives" there, i.e. can be detected/summed up with other PSPs. Disregarding all of the eventualities (see above):

What is the order of magnitude of the speed of passively transported ions in the dendritic tree?

• "I assume that ..." - why do you assume this? Commented Aug 28, 2017 at 15:41

Your edits do improve the question, but you are still asking two very different questions.

We don't really need to answer these questions:

How fast are ions transported passively inside the dendritic tree (and soma), probably depending on their atomic mass, charge, and of the local geometry?

What is the order of magnitude of the speed of passively transported ions in the dendritic tree?

Especially I want to know when a post-synaptic potential generated x μm away from the axon hillock "arrives" there, i.e. can be detected/summed up with other PSPs

The speed of propagation of electrical signals in neurons don't depend on particular masses of particular ions or diffusion per se. Instead, passive electrical conduction is best understood using the cable equations which treats conductance within neurons as electrical RC circuit. The electric field propagates at the speed of light.

Therefore, EPSPs actually propagate almost instantaneously (given how short the distances involved are relative to speed of light) in that they produce an electric field. However, the peak decays with distance, and the rise time of an EPSP depends on the compartment because of capacitive filtering along the length of the dendrite. The decay peak is described by the length constant:

$$\lambda = \sqrt{\frac{r_m}{r_i+r_o}}$$

In this equation, rm is the resistance of the membrane, ri is the resistance of the intracellular space, and ro is the resistance of the extracellular space and is effectively negligible.

The length constant is the distance at which the peak amplitude will have decayed 63% from the peak at the signal origin.

The temporal filtering comes from the capacitive properties of the membrane along the length of the dendrite. The authoritative works describing these properties are several papers in the 1960s from Rall, Rall, W. (1962). Theory of physiological properties of dendrites. Annals of the New York Academy of Sciences, 96(1), 1071-1092. and Rall, W. (1969). Time constants and electrotonic length of membrane cylinders and neurons. Biophysical Journal, 9(12), 1483-1508. are good references.

The effective time constant for signals originating closer to the soma is much faster because there is less membrane to charge along the way. Exactly "how far" you have to travel to make a certain amount of difference depends on the properties of the membrane, which vary across neurons, vary in different parts of the dendritic tree, and vary dynamically within a neuron due to synaptic transmission.

Magee, J. C. (2000). Dendritic integration of excitatory synaptic input. Nature reviews. Neuroscience, 1(3), 181. is a fairly accessible review that covers a lot of these issues and more. It's important to recognize that although passive propagation of signals is important in dendrites, it's never purely passive and active conductances have important roles as well; in many cells, these active conductances plus differences in the initial shape of EPSPs throughout the dendritic tree act to mitigate the effects of distance.

See also Koch, C., Rapp, M., & Segev, I. (1996). A brief history of time (constants). Cerebral cortex, 6(2), 93-101. for more on the time constant and some evolution of the ideas about neuronal time constants since Rall.

• Thanks. But alas, you destroyed my hope that the propagation of potentials (PSPs and action potentials) might be understandable by the flow of charged ions. Commented Sep 7, 2017 at 18:05
• Oh but the mathematics of electricity are much more tractable than Monte Carlo simulations of ion flow in a complex environment, so it isn't a hope at all but a dread. :) Commented Sep 7, 2017 at 19:22
• I'm having a horrid back spasm at the moment or I'd finish my answer but I'll just add in a comment that the things you actually care about for temporal integration are the durations of EPSPs, rather than their rise times. Commented Sep 7, 2017 at 19:23
• Will I find the number I was asking for (the order of magnitude of the speed by which EPSP signals are transported) in one of the papers you mention? Or do I have to calculate it solving complicated equations with some typical parameters? Alternatively, the effective speed can simply be defined as "(measured) distance of synapse and axon hillock" divided by "(measured) time until EPSP becomes effective at the axon hillock". Finally, this is what I am looking for. Commented Sep 8, 2017 at 9:29
• (The question arises, how this time can be measured. Maybe by stimulating simultaneously some synapses that have the same distance to the axon hillock and which are sufficient to evoke an action potential? Commented Sep 8, 2017 at 9:29