The Electrotonic Transformation: 
a Tool for Relating Neuronal Form to Function 
Nicholas T. Carnevale 
Department of Psychology, 
Yale University 
New Haven, CT 06520 
Brenda J. Claiborne 
Division of Life Sciences 
University of Texas 
San Antonio, TX 79285 
Kenneth Y. Tsai 
Department of Psychology 
Yale University 
New Haven, CT 06520 
Thomas H. Brown 
Department of Psychology 
Yale University 
New Haven, CT 06520 
Abstract 
The spatial distribution and time course of electrical signals in neurons 
have important theoretical and practical consequences. Because it is 
difficult to infer how neuronal form affects electrical signaling, we 
have developed a quantitative yet intuitive approach to the analysis of 
electrotonus. This approach transforms the architecture of the cell 
from anatomical to electrotonic space, using the logarithm of voltage 
attenuation as the distance metric. We describe the theory behind this 
approach and illustrate its use. 
1 INTRODUCTION 
The fields of computational neuroscience and artificial neural nets have enjoyed a 
mutually beneficial exchange of ideas. This has been most evident at the network level, 
where concepts such as massive parallelism, lateral inhibition, and recurrent excitation 
have inspired both the analysis of brain circuits and the design of artificial neural net 
architectures. 
Less attention has been given to how properties of the individual neurons or processing 
elements contribute to network function. Biological neurons and brain circuits have 
70 Nicholas Carnevale, Kenneth Y. Tsai, Brenda J. Claiborne, Thomas H. Brown 
been simultaneously subject to eons of evolutionary pressure. This suggests an essential 
interdependence between neuronal form and function, on the one hand, and the overall 
architecture and operation of biological neural nets, on the other. Therefore reverse- 
engineering the circuits of the brain appears likely to reveal design principles that rely 
upon neuronal properties. These principles may have maximum utility in the design of 
artificial neural nets that are constructed of processing elements with greater similarity 
to biological neurons than those which are used in contemporary designs. 
Spatiotemporal extent is perhaps the most obvious difference between real neurons and 
processing elements. The processing element of most artificial neural nets is essentially 
a point in time and space. Its activation level is the instantaneous sum of its synaptic 
inputs. Of particular relevance to Hebbian learning rules, all synapses are exposed to 
the same activation level. These simplifications may insure analytical and implementa- 
tional simplicity, but they deviate sharply from biological reality. Membrane potential, 
the biological counterpart of activation level, is neither instantaneous nor spatially 
uniform. Every cell has finite membrane capacitance, and all ionic currents are finite, 
so membrane potential must lag behind synaptic inputs. Furthermore, membrane 
capacitance and cytoplasmic resistance dictate that membrane potential will almost 
never be uniform throughout a living neuron embedded in the circuitry of the brain. 
The combination of ever-changing synaptic inputs with cellular anatomical and 
biophysical properties guarantees the existence of fluctuating electrical gradients. 
Consider the task of building a massively parallel neural net from processing elements 
with such "nonideal" characteristics. Imagine moreover that the input surface of each 
processing element is an extensive, highly branched structure over which approximately 
10,000 synaptic inputs are distributed. It might be tempting to try to minimize or work 
around the limitations imposed by device physics. Howevei', a better strategy might be 
to exploit the computational consequences of these properties by making them part of 
the design, thereby turning these apparent weaknesses into strengths. 
To facilitate an understanding of the spatiotemporal dynamics of electrical signaling in 
neurons, we have developed a new theoretical approach to linear electrotonus and a new 
way to make practical use of this theory. We present this method and illustrate its 
application to the analysis of synaptic interactions in hippocampal pyramidal cells. 
2 THEORETICAL BACKGROUND 
Our method draws upon and extends the results of two prior approaches: cable theory 
and two-port analysis. 
2.1 CABLE THEORY 
The modern use of cable theory in neuroscience began almost four decades ago with the 
work of Rall (1977). Much of the attraction of cable theory derives from the conceptual 
simplicity of the steady-state decay of voltage with distance along an infinite cylindrical 
cable: Y(x) = Vo e-x/ where x is physical distance and 2 is the length constant. This 
exponential relationship makes it useful to define the electrotonic distance X as the 
The Electronic Transfortnation.' A Tool for Relating Neuronal Fortn to Function 71 
logarithm of the signal attenuation: X = lnFo/V(x ). In an infinite cylindrical cable, 
electrotonic distance is directly proportional to physical distance: X = x/2. 
However, cable theory is difficult to apply to real neurons since dendritic trees are 
neither infinite nor cylindrical. Because of their anatomical complexity and irregular 
variations of branch diameter and length, attenuation in neurons is not an exponential 
function of distance. Even if a cell met the criteria that would allow its dendrites to be 
reduced to a finite equivalent cylinder (Rall 1977), voltage attenuation would not bear a 
simple exponential relationship to X but instead would vary inversely with a hyperbolic 
function (Jack et al. 1983). 
2.2 TWO-PORT THEORY 
Because of the limitations and restrictions of cable theory, Carnevale and Johnston 
(1982) turned to two-port analysis. Among their conclusions, three are most relevant to 
this discussion. 
Figure 1: Attenuation is direction-dependent. 
The first is that signal attenuation depends on the direction of signal propagation. 
Suppose that i and j are two points in a cell where i is "upstream" from j (voltage is 
spreading from i to j), and define the voltage attenuation from i to j: A(' = Vt/V  . Next 
suppose that the direction of signal propagation is reversed, so that j is now upstream 
from i, and define the voltage attenuation A( = Vj/V,. In general these two 
Jt 
attenuations will not be equal: A   A  
tj jt 
They also showed that voltage attenuation in one direction is identical to current 
attenuation in the opposite direction (Carnevale and Johnston 1982). Suppose current I i 
enters the cell at i, and the current that is captured by a voltage clamp at j is Ij, and 
define the current attenuation A I =Ii/Ij. Because of the directional reciprocity 
between current and voltage attenuation, A.{ = Ate. Similarly, if we interchange the 
tj gt 
current entry and voltage clamp sites, the current attenuation ratio would be A { = A . 
gt tj 
Finally, they found that charge and DC current attenuation in the same direction are 
identical (Carnevale and Johnston 1982). Therefore the spread of electrical signals 
between any two points is completely characterized by the voltage attenuations in both 
directions. 
72 Nicholas Carnevale, Kenneth Y. Tsai, Brenda J. Claiborne, Thomas H. Brown 
2.3 THE ELECTROTONIC TRANSFORMATION 
The basic idea of the electrotonic transformation is to remap the cell from anatomical 
space into "electrotonic space," where the distance between points reflects the 
attenuation of an electrical signal spreading between them. Because of the critical role 
of membrane potential in neuronal function, it is usually most appropriate to deal with 
voltage attenuations. 
2.3.1 The Distance Metric 
We use the logarithm of attenuation between points as the distance metric in electrotonic 
space: L = lnA/j (Brown et al. 1992, Zador et al. 1991). To appreciate the utility of 
this definition, consider voltage spreading from point i to point j, and suppose that k is 
on the direct path between i and j. The voltage attenuations are A, = V,/V, 
I/ A Vi/V j = Ai A/c:. This last equation and our definition of L 
Akj = Vk/V J , and = t/ 
establish the additive property of electrotonic distance LO.= Lik + Lkj. That is, 
electrotonic distances are additive over a path that has a consistent direction of signal 
propagation. This justifies using the logarithm of attenuation as a metric for the 
electrical separation between points in a cell. 
At this point several important facts should be noted. First, unlike the electrotonic 
distance X of classical cable theor)', our new definition of electrotonic distance L always 
bears a simple and direct logarithmic relationship to attenuation. Second, because of 
membrane capacitance, attenuation increases with frequency. Since both steady-state 
and transient signals are of interest, we evaluate attenuations at several different 
frequencies. Third, attenuation is direction-dependent and usually asymmetric. 
Therefore at ever), frequency of interest, each branch of the cell has two different 
representations in electrotonic space depending on the direction of signal flow. 
2.3.2 Representing a Neuron in Electrotonic Space 
Since attenuation depends on direction, it is necessary to construct transforms in pairs 
for each frequency of interest, one for signal spread away from a reference point (Vout) 
and the other for spread toward it (Vin). The soma is often a good choice for the 
reference point, but any point in the cell could be used, and a different vantage point 
might be more appropriate for particular analyses. 
The only difference between using one point i as the reference instead of any other point 
j is in the direction of signal propagation along the direct path between i and j (dashed 
arrows in Figure 2), where Vou t relative to i is the same as Ym relative to j and vice 
versa. The directions of signal flow and therefore the attenuations along all other 
branches of the cell are unchanged. Thus the transforms relative to i and j differ only 
along the direct path ij, and once the [/ut and [3'n transforms have been created for one 
reference i, it is easy to assemble the transforms with respect to any other referencej. 
The Electronic Transformation: A Tool for Relating Neuronal Form to Function 73 
Figure 2: Effect of reference point location on direction of signal propagation. 
We have found two graphical representations of the transform to be particularly useful. 
"Neuromorphic figures," in which the cell is redrawn so that the relative orientation of 
branches is preserved (Figures 3 and 4), can be readily compared to the original 
anatomy for quick, "big picture" insights regarding synaptic integration and 
interactions. For more quantitative analyses, it is helpful to plot electrotonic distance 
from the reference point as a function of anatomical distance (Tsai et al. 1993). 
3 COMPUTATIONAL METHODS 
The voltage attenuations along each segment of the cell are calculated from detailed, 
accurate morphometric data and the best available experimental estimates of the bio- 
physical properties of membrane and cytoplasm. Any neural simulator like NEURON 
(Hines 1989) could be used to find the attenuations for the DC Vou t transform. The DC 
Vin attenuations are more time consuming because a separate run must be performed for 
each of the dendritic terminations. However, the AC attenuations impose a severe com- 
putational burden on time-domain simulators because many cycles are required for the 
response to settle. For example, calculating the DC Vou t attenuations in a hippocampal 
pyramidal cell relative to the soma took only a few iterations on a SUN Sparc 10-40, but 
more than 20 hours were required for 40 Hz (Tsai et al. 1994). Finding the full set of 
attenuations for a Vin transform at 40 Hz would have taken almost three months. 
Therefore we designed an O(N) algorithm that achieves high computational efficiency 
by operating in the frequency domain and taking advantage of the branched architecture 
of the cell. In a series of recursive walks through the cell, the algorithm applies Kirch- 
hotTs laws to compute the attenuations in each branch. The electrical characteristics of 
each segment of the cell are represented by an equivalent T circuit. Rather than "lump" 
the properties of cytoplasm and membrane into discrete resistances and capacitances, we 
determine the elements of these equivalent T circuits directly from complex impedance 
functions that we derived from the impulse response of a finite cable (Tsai et al. 1994). 
Since each segment is treated as a cable rather than an isopotential compartment, the 
resolution of the spatial grid does not affect accuracy, and there is no need to increase 
the resolution of the spatial grid in order to preserve accuracy as frequency increases. 
This is an important consideration for hippocampal neurons, which have long mem- 
brane time constants and begin to show increased attenuations at frequencies as low as 2 
- 5 Hz (Tsai et al. 1994). It also allows us to treat a long unbranched neurite of nearly 
constant diameter as a single cylinder. 
Thus runtimes scale linearly with the number of grid points, are independent of 
frequency, and we can even reduce the number of grid points if the diameters of adjacent 
74 Nicholas Carnevale, Kenneth Y. Tsai, Brenda J. Claiborne, Thomas H. Brown 
unbranched segments are similar enough. A benchmark of a program that uses our 
algorithm with NEURON showed a speedup of more than four orders of magnitude 
without sacrificing accuracy (2 seconds vs. 20 hours to calculate the Vou t attenuations at 
40 Hz in a CA1 pyramidal neuron model with 2951 grid points) (Tsai et al. 1994). 
4 RESULTS 
4.1 DC TRANSFORMS OF A CA1 PYRAMIDAL CELL 
Figure 3 shows a two-dimensional projection of the anatomy of a rat CA1 pyramidal 
neuron (cell 524, left) with neuromorphic renderings of its DC Vou t and Vin transforms 
(middle and right) relative to the soma. The three-dimensional anatomical data were 
obtained from HRP-filled cells with a computer microscope system as described 
elsewhere (Rihn and Claiborne 1992, Claiborne 1992). The passive electrical properties 
used to compute the attenuations were R i = 200 cm, C m = 1 xF/cm 2 (for nonzero 
frequencies, not shown here) and R m = 30 k.Qcm 2 (Spruston and Johnston 1992). 
Om 
Figure 3:CA1 pyramidal cell anatomy (cell 524, left) with neuromorphic 
renderings of Vou t (middle) and Vt. n (right) transforms at DC. 
The Vou t transform is very compact, indicating that voltage propagates from the soma 
into the dendrites with relatively little attenuation. The basilar dendrites and the 
terminal branches of the primary apical dendrite are almost invisible, since they are 
nearly isopotential along their lengths. Despite the fact that the primary apical dendrite 
has a larger diameter than any of its daughter branches, most of the voltage drop for 
somatofugal signaling is in the primary apical. Therefore it accounts for almost all of 
the electrotonic length of the cell in the Vou t transform. 
The Vin transform is far more extensive, but most of the electrotonic length of the cell is 
now in the basilar and terminal apical branches. This reflects the loading effect of 
downstream membrane on these thin dendritic branches. 
4.2 SYNAPTIC INTERACTIONS 
The transform can also give clues to possible effects of electrotonic architecture on 
voltage-dependent forms of associative synaptic plasticity and other kinds of synaptic 
interactions. Suppose the cell of Figure 3 receives a weak or "student" synaptic input 
The Electronic Transformation: A Tool for Relating Neuronal Form to Function 75 
located 400 xm from the soma on the primary apical dendrite, and a strong or "teacher" 
input is situated 300 xm from the soma on the same dendrite. 
[] student 
 teacher 
200 !jj.n  L= 1 . 
A. cell 524 B. cell 503 
Figure 4: Analysis of synaptic interactions. 
The anatomical arrangement is depicted on the left in Figure 4A ("student" = square, 
"teacher" = circle). The I/in transform with respect to the student (right figure of this 
pair) shows that voltage spreads from the teacher to the student synapse with little 
attenuation, which would favor voltage-dependent associative interactions. 
Figure 4B shows a different CAI pyramidal cell in which the apical dendrite bifurcates 
shortly after arising from the soma. Two teacher synapses are indicated, one on the 
same branch as the student and the other on the opposite branch. The Vin transform 
with respect to the student (right figure of this pair) shows clearly that the teacher 
synapse on the same branch is closely coupled to the student, but the other is electrically 
much more remote and less likely to influence the student synapse. 
5. SUMMARY 
The electrotonic transformation is based on a logical, internally consistent conceptual 
approach to understanding the propagation of electrical signals in neurons. In this 
paper we described two methods for graphically presenting the results of the 
transformation: neuromorphic rendering, and plots of electrotonic distance vs. 
anatomical distance. Using neuromorphic renderings, we illustrated the electrotonic 
properties of a previously unreported hippocampal CA1 pyramidal neuron as viewed 
from the soma (cell 524, Figure 3). We also extended the use of the transformation to 
the study of associative interactions between "teacher" and "student" synapses by 
analyzing this cell from the viewpoint of a "student" synapse located in the apical 
dendrites, contrasting this result with a different cell that had a bifurcated primary 
apical dendrite (cell 503, Figure 4). This demonstrates the versatility of the electrotonic 
transformation, and shows how it can convey the electrical signaling properties of 
neurons in ways that are quickly and easily comprehended. 
This understanding is important for several reasons. First, electrotonus affects the 
integration and interaction of synaptic inputs, regulates voltage-dependent mechanisms 
of synaptic plasticity, and influences the interpretation of intracellular recordings. In 
addition, phylogeny, development, aging, and response to injury and disease are all 
accompanied by alterations of neuronal morphology, some subtle and some profound. 
76 Nicholas Carnevale, Kenneth Y. Tsai, Brenda J. Claiborne, Thomas H. Brown 
The significance of these changes for brain function becomes clear only if their effects 
on neuronal signaling are reckoned. Finally, there is good reason to expect that 
neuronal electrotonus is highly relevant to the design of artificial neural networks. 
Acknowledgments 
We thank R.B. Gonzales and M.P. O'Boyle for their contributions to the morphometric 
analysis, and Z.F. Mainen for assisting in the initial development of graphical 
rendering. This work was supported in part by ONR, ARPA, and the Yale Center for 
Theoretical and Applied Neuroscience (CTAN). 
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