|input impedance ZN||(local voltage change)/(local current injection)|
|transfer impedance Zc||(local voltage change)/(remote current injection)
(remote voltage change)/(local current injection)
|voltage transfer ratio k||(voltage downstream)/(voltage upstream)
Identical to current and charge transfer ratio in the opposite direction.
|voltage attenuation A||(voltage upstream)/(voltage downstream)
Identical to current and charge attenuation in the opposite direction.
|electrotonic distance L
(see Note below)
|natural log of A|
NOTE: The electrotonic distance computed by NEURON is defined by attenuation, but the classical definition is (anatomical distance/length constant). These two measures of electrotonic length are identical for an infinite cylindrical cable. However, the measure computed by NEURON always has a simple, direct relationship to attenuation, regardless of cellular anatomy, whereas the classical measure only has meaning in cells that meet several very specific constraints (such as the "3/2 power branching criterion"), and even then it does not have a simple relationship to attenuation. The new definition of electrotonic distance also preserves the direction-dependence of attenuation, which the classical definition obscures.
neurondemoand select the Pyramidal cell model. Examine the side view of the anatomy of the cell in the Shape plot. Rotate, zoom in, and check it out from different vantage points (if you have any questions, here's how).
NEURON's tools for electrotonic analysis are gathered into four different "styles":
They are accessible through NEURON Main Menu / Tools / Impedance. In this exercise you will start to learn how to use each of them. To save screen space, close a tool when you are done with it.
The Frequency tool can be used to study electrical coupling between any two points in a cell. Suppose an interesting signal is generated at some location in a neuron, e.g. by a synapse or by active conductances. An electrode is attached to the cell, but not necessarily where the signal is produced. The electrode may be used with a current clamp to inject current and record fluctuations in membrane potential Vm, or with a voltage clamp that records the clamp current Ic that is needed to regulate Vm.
This experimental situation brings several questions to mind, such as
Bring up the Frequency tool: NEURON Main Menu / Tools / Impedance / Frequency
|Top panel controls operation of the tool, and tells what is happening. Note
Middle panel shows anatomy of the cell, indicating where current is injected (blue dot) and where Vm is measured (red dot). Zoom in or out as needed to see the whole cell.
Bottom panel in this example shows natural log of voltage attenuation from the injection site to the measurement site as a function of frequency over the range 0.1 to 1000 Hz. For most cells, attenuations at DC and 0.1 Hz are nearly identical.
Things to do:
The Path tool is useful when the general location of the signal source is known, or when there are several independent signal sources at different locations. It performs the same kind of analyses that the Frequency tool does, but it allows the user to examine signaling between an electrode and a region instead of a specific location.
Bring up the Path tool: NEURON Main Menu / Tools / Impedance / Path
Top panel. Note particularly
Middle panel. Click on the Shape Plot to set the location of the electrode (red dot), the path start (blue dot), and the path end (green dot). The direct path between the start and end is highlighted in red. Anatomical distance along this path is the independent variable against which analysis results of are plotted.
Bottom panel. For all points on the path, shows natural log(A) for voltage spreading toward (Vin) or away from (Vout) the electrode, according to the choice set in the Top panel. The path start is on the left of the horizontal axis, and the end is on the right. If the electrode is located on the path, this is the distance from the electrode in um; if the electrode is not on the path, this is the distance to the location on the path that is closest to the electrode.
Things to do:
This tool is the ultimate extension of the approach used by the Path tool: it shows the log of voltage attenuation for each point in the cell relative to the electrode or reference point.
Bring up the log(A) vs x tool: NEURON Main Menu / Tools / Impedance / log(A) vs x
Top panel. The only significant difference from the Path tool is the button
labeled Shape Select Action. This button enables two important operations
in the Shape Plot (middle panel). The first is "Move electrode" which lets you
move the electrode to a new location by clicking on a neurite.
The second is "Show Position" which helps you discover the mapping from
the Shape Plot to the log(A) vs x plot:
click on a neurite to see both it and the corresponding line
in the log A vs. x plot turn red.
The bottom panel's graph menu has a "Show position" item that
does the same thing in the opposite direction.
Middle panel. Since attenuation is computed over the entire cell, the only location the user specifies is the position of the electrode (red dot).
Bottom panel. For every section throughout the cell, this panel shows ln(A) for voltage spreading toward (Vin) or away from (Vout) the electrode. The abscissa is the distance in um along the direct path from the soma (not the electrode) to each point. To discover which neurite corresponds to a line in this graph, click on the menu box (square in upper left corner of this graph) and select the "show position" item. Then click on a line to see it and the corresponding neurite turn red.
Things to do:
Leave the electrode at the soma.
Follow these steps to discover passive normalization for yourself!
But synapses aren't voltage sources. They're much more like current sources. In other words, a synapse would deliver nearly the same current to a neuron regardless of where it is attached to the cell. Therefore voltage transfer ratio in the Vin direction (from synapse to soma) does not predict the relationship between synaptic efficacy and synaptic location. Instead, the best predictor of synaptic efficacy is normalized transfer impedance. This is identical to the voltage transfer ratio in the Vout direction!
So just click on the Vout radio button and you see that a synapse attached to a basilar dendrite will produce nearly the same somatic PSP no matter how far it is from the soma! This is the phenomenon that David Jaffe and I call passive normalization : variation of somatic PSP amplitude with synaptic distance is reduced ("normalization"), and it doesn't require active currents to happen ("passive"). For more information, see our paper.
Perhaps the most intuitive representation of electrotonic architecture is to redraw the branched anatomy of the cell in a way that preserves the relative orientation of the branches, using line segments that are proportional to natural log(A) between adjacent points instead of the anatomical branch lengths. These neuromorphic renderings of the electrotonic transform warp the anatomy of the cell so that the proximity of points to each other is a direct indication of the degree of electrical coupling between them: tightly coupled points appear close to each other, and points that are electrically remote from each other are shown farther apart. The overall form of a neuromorphic figure parallels cellular anatomy, so it is easy to identify structural features of the cell, such as basilar or apical dendrites and particular dendritic segments or branch points.
Bring up the Shape tool: NEURON Main Menu / Tools / Impedance / Shape
Top panel. The controls for the Shape tool are very simple. Because of the
direct visual parallels between the Shape plot (middle panel) and the form
of the neuromorphic rendering (bottom panel), there is no need for special
functions to demonstrate the correspondence between lines in these two panels.
There is no Plot button because there is no way to represent
Zin or Ztransfer by changing branch lengths in the neuromorphic figure.
Middle panel. This shows the anatomy of the cell and the location of the electrode or reference point (red dot), as in the log(A) vs x style.
Bottom panel. This displays the neuromorphic rendering of one of the components of the electrotonic transform. The distance of a point from the site of the electrode is proportional to the natural logarithm of attenuation for voltage spreading toward (Vin) or away from (Vout) the electrode, according to the selection in the Top panel. The calibration bar represents one log unit of attenuation, i.e. the distance that signifies an e-fold decay of voltage.
Things to do:
Leave the electrode at the soma.
Carnevale, N.T., Tsai, K.Y., Claiborne, B.J., and Brown, T.H. The electrotonic transformation: a tool for relating neuronal form to function. In: Advances in Neural Information Processing Systems, vol. 7, edited by G. Tesauro, D.S. Touretzky, and T.K. Leen. Cambridge, MA: MIT Press, 1995, p. 69-76. Posted in html format at http://www.neuron.yale.edu/static/papers/NIPS94/nipsfin.html or see this local copy.
Jaffe, D.B. and Carnevale, N.T. Passive normalization of synaptic integration influenced by dendritic architecture. Journal of Neurophysiology 82:3268-3285, 1999. Preprint available from http://www.neuron.yale.edu/static/papers/jnp99/pasnorm.pdf