The Morphoelectrotonic Transform

A Mathematica notebook demonstrates a graphical approach to dendritic function.

by Hagai Agmon-Snir

Neurons are basic elements of the nervous system. A neuron typically consists of a cell body, or soma and projective structures usually classified as axons and dendrites. Dendrites are the branching ("tree-like") processes that receive and integrate signals. See for example [Bloom and Lazerson 1988]. 

This sample notebook shows the uses of functions developed to help elucidate the functional significance of dendrites, using a simple graphical approach that offers an intuitive understanding of electrotonic structure. The theory is explained in detail in [Zador, Agmon-Snir and Segev 1995]. Briefly, our approach is based on two novel measures of dendritic electrotonic structure, the log-attenuation (Subscript[L, ij]) (Zador, Claiborne and Brown 1991) and the propagation delay (Subscript[P, ij])(Agmon-Snir and Segev 1993). (Subscript[L, ij]) measures the efficacy and (Subscript[P, ij])the speed of signal transfer between two points i and j. The measures are additive, in the sense that if j lies between i, and k, then (Subscript[L, ik]) = (Subscript[L, ij]) + (Subscript[L, jk]) and (Subscript[P, ik]) = (Subscript[P, ij]) + (Subscript[P, jk]). This property serves as the basis for the morphoelectrotonic transform (MET), a graphical mapping from morphological into electrotonic space.

The package on which this notebook is based contains functions other than the ones demonstrated here. Readers who are interested in further experimentation are encouraged to contact the author. The morphological files used are variations of the Eutectic format (see acknowledgement). The readDouglasTree and the readEutecticTree functions can be modified to other formats. The header of the files was cut and an "end" sign was put at the end, but one can, of course, change it.

The mathematical theory behind the dendritic delay is elaborated in [Agmon-Snir 1995]. A popular and elementary textbook that explains the MET is [Johnston and Wu 1995].

Loading the MET package and reading the dendrite file

Needs["MET`", "MET.m"]

The sample file is a reconstruction of a layer 5 pyramidal cell from the cat visual cortex (Douglas, Martin and Whitteridge 1991). It consists of a relatively short bushy basal tree, with fine processes terminating about 200 μm from the soma, and a bifurcating apical tree with fine processes terminating about 1000 μm from the soma.

readDouglasTree["pyramid5"];

readDouglasTree reads the R. Douglas files (with minor changes as explained above) into Mathematica. It assumes that coordinates and diameters are in 0.1 microns. In the Mathematica 
program, all the values are saved in cm.

convert2InternalTree;

convert2InternalTree converts the data to the internal tree data structure.

somaIsDendnum;

In reconstructing a neuron, one of the 'dendrites' is the outline of the soma. somaIsDendnum is the number of this 'dendrite'."

dendnum;

dendnum is the number of dendrites reconstructed (including the soma outline).

Showing the original structure of the dendrite

The package contains functions for color as well as black-and-white graphics (see cover of this issue). The neurons and the transformed neurons can be shown with or without a relative scale bar. Here the neuron is shown with a 100 micron bar.

makeGraphicsCommandBW2D

showTheGraphWithBar[0.01];

Biophysical calculations and examples of transforms

spinefactor = 3/2 means the dendritic spines add 50% to the area. In this sample notebook, spinefactor is the same for all the dendrites, but one can change it, of course. rm and ri are in ohms. epsilon is the ratio between the membrane time constant of the soma and the membrane time constant of the dendrites. The algorithm can treat the case of different rm, ri, cm and spinefactors in different branches. It is just the input mechanism in this sample notebook that is restricted to one set of biophysical values across the tree.

spinefactor=3/2;rm=20000.0/spinefactor//N;ri=100.0;
cm=1.0 spinefactor//N;epsilon=1.0;tau=rm cm/1000.0;

initialBiophysicalCalculations

The following sequences of package functions give us an "attenogram" for the cases of voltage output and voltage input and a "delayogram" for voltage input.

pointcalc[MET`Private`soma,{}];

makeAttenuationTransform;

makeMETGraphicsCommandBW2D

showTheGraphWithBar[0.1];

(For proper calculation of the voltage-in case, "vinpointcalc" must be called after "pointcalc" with the same argument.)

pointcalc[MET`Private`soma,{}];

vinpointcalc[MET`Private`soma,{}];

makeVinAttenuationTransform;

makeMETGraphicsCommandBW2D

showTheGraphWithBar[1];

makeVinDelayTransform;

makeMETGraphicsCommandBW2D

showTheGraphWithBar[10];

More transforms

We have illustrated a small sample of the functions available in the met.m package. Each MET emphasizes a different aspect of electrotonic structure, yet consideration of just a few can provide a rather complete functional view of a neuron's tree. Applying the MET to different neuron types (layer 5 and layer 2/3 cortical pyramidals, hippocampal pyramidals, and cerebellar Purkinje), we gain insights into the different integrative properties of these neurons. We can also show that the MET can be extended to treat ionic signaling in dendrites and spines.

Acknowlegement

The morphological files used are variations of the Eutectic format (Eutectic Electronics, Inc., 919-782-3000). We thank Rodney Douglas for making his neuron reconstructions available.

The author is indebted to the editor of this journal for his help in adapting the Mathematica notebook to the journal style.

References

Agmon-Snir, H. A novel theoretical approach to the analysis of dendritic transients. Biophys. J. 69: 1633-56, 1995

Agmon-Snir, H. and Segev, I., Signal delay and input synchronization in passive dendritic structures. Journal of Neurophysiology 70:2066-2085, 1993.

Bloom, F. E., Lazerson, A., Brain, Mind and Behavior. W. H. Freeman, 1988.

Douglas, R., Martin, K., and Whitteridge, D., An intracellular analysis of the visual responses of neurones in cat visual cortex. Journal of Physiology (London) 440: 659-696, 1991. 

Johnston, D. and Wu, S. M., Foundations of Cellular
Neurophysiology, MIT Press, 1995.

Zador, A., Agmon-Snir, H., and Segev, I. 1995. The morphoelectrotonic transform: a graphical approach to dendritic function. Journal of Neuroscience 15: 1669-1682, 1995. 

Zador, A., Claiborne, B., and Brown, T., Electrotonic transroms of hippocampal neurons. Soc. Neuroscience Abstracts 17: 1515, 1991.

About the Author

Hagai Agmon-Snir is a researcher with the Mathematical Research Branch,
National Institutes of Health. His research interests include computational
neuroscience, especially the role of dendrites in neuronal computation.

Hagai Agmon-Snir                                    
Mathematical Research Branch, NIH
9190 Rockville Pike - Suite 350
Bethesda, MD 20814-3800
USA   
Tel: (301) 496-6136, (301) 496-4325
Fax: (301) 402-0535
E-mail: hagai@helix.nih.gov