# Rall model

Wilfrid Rall (2009), Scholarpedia, 4(4):1369. | doi:10.4249/scholarpedia.1369 | revision #91692 [link to/cite this article] |

The designation, **Rall model**, usually refers to one or more of several closely related biophysical-mathematical models of neurons that have significant dendritic trees. The importance of the cable properties of dendrites was emphasized in 1957, when Wilfrid Rall corrected
erroneous estimates of the membrane time constant of motoneurons in cat spinal cord.
During the 1960s, cable models (a cylinder of finite length, and a ten-compartment model)
were used to predict experimental results that were successfully tested with research
collaborators at the National Institutes of Health. This research established the significance of synaptic input to distal
dendritic locations; it also explored properties of dendritic spines, and predicted dendro-dendritic synapses in the olfactory bulb.

## Overview

A conceptual overview of these models can be provided by referring to Figure 1. This shows an extensively branched dendritic tree. Because all of the branches are assumed to be cylinders of uniform passive membrane, the distribution of membrane potential along the length of every branch must obey the cable equation (Eq.(1), below). Also, the branching in Figure 1 is assumed to be symmetrical, and when the branch diameters satisfy a particular (d^{3/2}) constraint, it was found (Rall, 1962, 1964) that dendritic locations in this tree can be mapped to an equivalent cylinder, as indicated schematically in Figure 1. For injection of current to the tree-trunk, the spread of current and the spreading change in membrane potential from the trunk into all of the branches, correspond to a mathematical solution of the cable equation in the equivalent cylinder. Thus, the equivalent cylinder, of finite length, proved to be a valuable reduced biophysical-mathematical model of the idealized dendritic tree. With this model, it was possible to obtain analytical solutions to a number of neurophysiologically interesting mathematical boundary value problems (Rall 1962), as illustrated and discussed below, with Eq.(2) and Figure 2.

For computational purposes, it was found advantageous to use a chain of ten equal compartments to approximate the equivalent cylinder, as indicated in Figure 1. Here, the partial differential equation for the cylinder is replaced by a system of ordinary differential equations for the chain of compartments (Rall 1964). Compartment-1 can be viewed as the neuron soma, and compartments 2 to 10 represent increasing distance out into the dendritic tree. Here, synaptic excitation and/or synaptic inhibition can be specified to occur in particular compartments to explore the consequences of different input locations and different spatiotemporal patterns of synaptic input (Rall 1964).

It is noted that the simplification of collapsing distal branching was avoided in a later mathematical treatment (Rall & Rinzel 1973; Rinzel & Rall 1974) which provided explicit analytical solutions for input delivered to a single branch; see later section titled: 7 Solutions for input to a single branch of an idealized dendritic tree, below.

The circuit diagram, at the upper right of Figure 1, shows the biophysical membrane model that was used. It consists of a membrane capacitance (per unit area), which is in parallel with three membrane conductance (per unit area) pathways. Subscript (r) identifies the resting membrane conductance, which is in series with the resting membrane battery. Subscript (e, or epsilon) identifies the variable synaptic excitatory conductance, which is in series with the excitatory battery; this conductance is zero under resting conditions, but is turned on by an excitatory synaptic input. Subscript (j) identifies the variable synaptic inhibitory conductance, which is in series with the inhibitory battery; this conductance is zero under resting conditions, but is turned on by inhibitory synaptic input. These conductance pathways correspond to models proposed by Fatt & Katz (1953) and Coombs, Eccles & Fatt (1955), following upon earlier insights by Hodgkin & Katz (1949), Fatt & Katz (1951) and Hodgkin & Huxley (1952).

By specifying membrane excitability in a compartmental model of a mitral cell, it was possible to simulate antidromic propagation of an impulse along axonal compartments to activate the soma and dendritic compartments of this model. When this compartmental computation was combined with a model of a mitral cell population having spherical cortical symmetry (Rall & Shepherd 1968), it was found possible to simulate spatiotemporal, extracellular field potentials that had been previously recorded in the olfactory bulb of rabbit. A brief account of how this modeling led to a prediction of dendro-dendritic synapses is provided in two later sections titled: 5 Mitral and granule cell dendrites: olfactory bulb model, and 6 Dendro-dendritic synapses predicted and found.

Later modeling of synaptic input to dendritic spines provided useful insights about how such spines could be involved in neuronal plasticity, and in integrative and logical processing in distal dendrites, as noted in a later section titled: 8 Dendritic spine model and clusters of spines on branches.

## d^{3/2} Constraint on branch diameters

When exploring steady state solutions for a non-symmetric tree, where branch diameters and lengths could have arbitrary values (Rall 1959), it became clear, because the input conductance of each branch cylinder depends on the 3/2 power of its diameter, that a special case occurs at those branch points where the two daughter branches have diameters that satisfy the following constraint: the sum of their 3/2 power values is equal to the 3/2 power of the parent branch diameter. When this constraint is satisfied at all branch points of a tree, this tree can be mapped to an equivalent cylinder (provided also that all terminal branches end at the same electrotonic distance from the trunk). For symmetric branching, this means that each daughter branch diameter equals 0.63 of the parent branch diameter. (To see this intuitively, note that 0.8 is the square root of 0.64, so the 3/2 power of 0.64 equals the cube of 0.8, namely 0.512; similarly, the 3/2 power of 0.63 equals 0.50). A useful approximate example of such a symmetric tree has a trunk diameter of 10 microns, with successive daughter branch diameters of 6.3, 4.0, 2.5, 1.6, and 1.0 microns. It is noteworthy that measurements of branch diameters in motoneurons of cat spinal cord have found rough agreement with the d^{3/2} constraint; see review article by Rall et al., (1992) for examples. When there is consistent deviation from this constraint, one can make use of a more general partial differential equation (Rall 1962a) that corresponds to an equivalent taper; this has been explored in Goldstein & Rall (1974); see also Holmes & Rall (1992).

## Equivalent cylinder model and cable equation

The first model was a cylinder of finite length, consisting of uniform passive nerve membrane (Rall, 1962a). In this reduced model, a dendritic tree was collapsed into a single cylinder; the neuron soma lies at one end, \(X=0\ ,\) while the dendritic terminals lie at the other end, \(X=L\ .\) Because passive membrane is assumed, the spatiotemporal distribution of membrane potential along the cylinder must obey a partial differential equation, known as the **cable equation** (a derivation is provided in Rall 1977, pages 64-67); this can be expressed
\[\tag{1}
\frac{\partial V}{\partial T} = - V + \frac{\partial^2 V}{\partial X^2}
\]

where \(V = V_m - E_r\) represents the departure of the membrane potential, \(V_m\ ,\) from its resting value, \(E_r\) (and \(V_m\) is intracellular voltage \(V_i\) minus extracellular voltage \(V_e\)). Also, \[X=x/\lambda\ ,\] with \(\lambda = \sqrt{r_m/r_i} = \sqrt{(R_m/R_i)(d/4)}\) \[T=t/\tau_m\ ,\] with \(\tau_m = r_mc_m = R_mC_m\ .\) Here, \(d\) is the cylinder diameter, while \(r_i\) is the intracellular (core) resistance per unit length of the cylinder and \(c_m\) and \(r_m^{-1}\) are the membrane capacity and membrane conductance, respectively, per unit length of the membrane cylinder; \(R_m\) and \(C_m\) apply to unit membrane area, while\(R_i\) is the volume resistivity of the intracellular medium.

For "sealed ends", meaning no current flows out of either end, the mathematical boundary conditions are \(\partial V/\partial X = 0\ ,\) at both \(X = 0\) and \(X = L\ .\) Using the classical method known as separation of variables, a general solution of this boundary value problem can be expressed \[\tag{2} V(X,T) = \sum_{n=0}^\infty B_n \cos(\alpha_n X) e^{-(1+\alpha_n^2)T} \]

where \(n\) is any positive integer, or zero, and \[\tag{3} \alpha_n = n\pi/L \]

The values of \(\alpha_n^2\) are known as the eigenvalues of the boundary value problem. The \(B_n\) are constants to be determined from the initial condition.

A related boundary value problem distinguishes between two regions of the cylinder, corresponding to proximal and distal regions of the dendritic tree, where one region has a uniform synaptic excitatory input turned on for a short time. Once this synaptic input was turned off, the whole cylinder obeys Eq.(2). Computations using these analytical solutions demonstrated how the transient shape of a synaptic potential (EPSP at the soma) depends upon distal versus proximal locations of synaptic input: for distal input, the EPSP rises more slowly to a later and broader peak (of lesser amplitude), compared to an earlier, sharper peak for proximal dendritic input locations (Rall, 1962a), as illustrated in Figure 2.

It is noteworthy that when the expression (3) is introduced into the theoretical expression for the exponent in Eq.(2), we obtain an expression that implies a set of time constants, \(\tau_n\) (for the passive cylinder with sealed ends), as follows \[\tag{4} \tau_0/\tau_n = 1 + (n\pi/L)^2 \]

Note that \(\tau_0=\tau_m\) governs the decay of a uniformly distributed membrane potential, while the \(\tau_n\ ,\) for \(n>0\ ,\) have been called **equalizing time constants**, because they govern the more rapid decay of non-uniformly distributed components of the membrane potential. Because Eq.(4) can be rearranged to provide the expression
\[\tag{5}
L = \frac{n\pi}{\sqrt{\tau_0/\tau_n-1}}
\]

one can see that the value of \(L\) can be calculated from the ratio of two time constants. This has provided a very useful method for estimating the electrotonic length, \(L\ ,\) from analysis of experimental transients, as discussed and illustrated in Rall (1969).

## Compartmental model

The next model was a chain of ten equal compartments (Rall, 1964), see Figure 1. This can be regarded as an approximation to the cylinder of finite length. Here compartment-1 represents the neuron soma, while compartments-2 through 10 represent increasing distance out into the dendritic tree. The continuous variation of \(V\) with \(X\) in the first model is replaced by stepwise differences in \(V\) between adjacent, connected regions of lumped membrane. This model offers valuable computational advantages. In 1962, Rall made use of a computer program, SAAM, that had been developed at NIH by Mones Berman for studies in metabolic kinetics, to solve the equations. Today computer programs designed for neuroscience, such as NEURON, developed by Hines, and GENESIS, developed by Wilson and Bower are used for this purpose.

It became possible to compute EPSP shapes for many different dendritic locations of synaptic input (Rall, 1964), as illustrated in Figure 3, for square (on-off) synaptic conductances.

Later such computed EPSP shapes were refined by using a transient time course to govern the synaptic conductance, in order to enhance the comparison with experimental EPSP shapes (Rall, 1967, Rall et al, 1967). For each EPSP shape, two shape index values, time to peak and half-width, could be used to construct two-dimensional shape index plots, leading to theoretical shape index loci (for different input locations and for different synaptic input transients); see Figure 4.

These theoretical loci provided explicit theoretical predictions that could be compared with experimental EPSP shapes. A successful NIH research collaboration with Bob Burke, Tom Smith, Phil Nelson & K. Frank, (Rall et al 1967) persuaded most neurophysiologists about the significant contribution of dendritic synapses in motoneurons of cat spinal cord (after years of denial by Eccles). The agreement of experimental EPSP shapes with theoretical predictions was further demonstrated by Julian Jack and colleagues at Oxford (Jack et al, 1971), by Mendell and Henneman (1971) at Harvard, and by Iansek and Redman (1973) in Australia. Later a remarkable experiment by Redman and Walmsley (1983), in Australia, succeeded in combining electrophysiology and histology (in the identical neuron) to show agreement between two actual known synaptic input locations (for two individual afferent fibers) and the theoretical input locations implied by the two different recorded EPSP shapes.

This compartmental model was also used to demonstrate the significantly different results obtained when comparing/contrasting spatiotemporal sequences of synaptic input (Rall, 1964); see Figure 5. Such differences could have relevance for pattern discrimination and for movement detection.

Different computations explored nonlinear interactions between synaptic excitation and synaptic inhibition (Rall, 1962, 1964). Synaptic inhibition was shown to be less effective when it was located distal to the synaptic excitation; it was almost equally effective when placed in the same compartment as the synaptic excitation or at the soma compartment.

With regard to more general compartmental models, it must be emphasized that the ten compartment model, used above, is a very simple, passive special case. Compartmental models can be designed to go beyond the constraint of passive membrane properties and unbranched cylinders. Different non-linear and voltage dependent membrane properties can be specified for any compartment (e.g. the model in the next section below); compartments can be of different size, and also, dendritic branching can be represented explicitly (Rall, 1964).

## Mitral and granule cell dendrites: olfactory bulb model

A more complicated model, combining compartmental and cortical modeling, was used by Rall and Shepherd (1968) to simulate the spatiotemporal extracellular (field) potentials that had been recorded in experiments with the olfactory bulb of rabbit by (Phillips, Powell & Shepherd, 1963). This involved synchronous antidromic activation of a very large population of mitral cells, arranged in an almost spherically symmetrical cortex, with their primary dendrites aligned radially outward. To model mitral cell activation, a compartmental model was used; this had three small axonal compartments, and a larger soma compartment, all of which had specified active membrane properties. (Note that these active properties depended on a new mathematical model that was designed to generate conductance transients similar to those of the Hodgkin-Huxley model; this was computationally economical; also the values of the H-H parameters were not known for neurons of rabbit). These active properties provided propagation of an antidromic action potential from the axonal compartments to the soma. The dendritic compartments had passive membrane in some computations, and active membrane in other computations. Because of the large population of simultaneously activated mitral cells in the olfactory bulb, this population was idealized as a spherically symmetrical cortical layer in which the extracellular current generated by the mitral cells flows radially, producing spherical equipotential contours whose transient values could be computed. However, because the symmetry of the actual olfactory bulb is not perfect, and because it is "punctured" by the lateral olfactory tract, it was necessary to consider a small secondary extracellular current whose extra-bulbar path includes the reference electrode. Thus the computational model included a "potential divider" correction, which resulted in agreement with the experimental data; see Rall and Shepherd (1968) for the details.

There is also a very large population of granule cells whose dendrites intermingle with dendrites of the mitral cells in the external plexiform layer (EPL) of the olfactory bulb; the granule cells also extend dendrites into a deeper layer. A granule cell was also represented by a chain of compartments; these compartments had passive membrane and they were assumed to extend from the EPL, deep into the granular layer (GRL) of the olfactory bulb. Computations with this granule cell model also required a "potential divider" correction, to simulate the effect of a distant reference electrode.

## Dendro-dendritic synapses predicted and found

The successful simulation of the experimental field potentials implied that the granule cell population must receive synaptic excitatory input within the external plexiform layer (EPL) of the bulb, at the very time that the mitral secondary dendrites are depolarized in the EPL. This suggested that the depolarized mitral cell dendrites must deliver synaptic excitation to the granule cell dendrites in the EPL; this would imply *dendro-dendritic* excitatory synaptic contacts, which were not yet known to exist. Also, because the mitral cells were subsequently inhibited, this suggested that synaptic inhibition must be delivered by the granule cell dendrites to the mitral cell dendrites in the EPL; this implies dendro-dendritic inhibitory synaptic contacts, which were not yet known to exist. Only seven months later (March 1965), both kinds of dendro-dendritic synapses were found by Tom Reese and Milton Brightman in their independent electron-microscopic research, at NIH. Because the theoretical model and the experimental results agreed so well, a joint paper was prepared and submitted to Science in 1965; their referee found it not to be of general interest. The paper was later accepted for publication in Experimental Neurology (Rall et al, 1966). Dendro-dendritic synapses were controversial for a while, but they have since been found in other regions of the CNS.
It should be noted that these synapses provide a new pathway for lateral inhibition; also such synapses provide for graded interactions that do not make use of an all-or none impulse.

## Solutions for input to a single branch of an idealized dendritic tree

An enriched model was used to explore the consequences of input to a single branch of an extensively branched dendritic tree. By using superposition methods, the analytical solutions of the cylinder model (see earlier section titled: 3 Equivalent cylinder model and cable equation above) were combined to provide solutions for the case of idealized branching; the branching was assumed symmetric, with branch diameters that satisfy the d^{3/2} constraint. This neuron model allowed for several equal dendritic trees, \(N\) in number; each tree was equivalent to a cylinder of electrotonic length, \(L\ ,\) and had \(M\) orders of symmetric branching. For input to a single branch of one of these trees, steady state solutions (Rall & Rinzel, 1973), and transient solutions (Rinzel & Rall, 1974), were provided, illustrated and discussed. Insights were provided regarding input resistance and impedance values at different input locations in the tree, as well as voltage attenuation in different branches of the tree.

## Dendritic spine model and clusters of spines on branches

One reason to explore solutions for input to a single branch was an interest in dendritic spines, together with the question: when a spine head receives a synaptic input, how much of the potential generated in the spine head membrane is delivered to the tree-branch where the usually narrow (high resistance) spine stem is attached? It was found important to consider the ratio of spine-stem resistance to the input resistance of the tree-branch (Rall & Rinzel, 1971). For a ratio of 0.01 or less, steady input to the spine head is delivered to the branch without significant attenuation. For a ratio of 100 or more, the voltage delivered to the branch is negligible. For an intermediate range, from 0.1 to 10, steady voltage attenuation ranges approximately between 10% and 90%. This was identified as an "operating range" for possible synaptic plasticity and learning.

Related models were used to explore the effects of excitable spines (Miller et al. 1985), and of clusters of spines (both passive and excitable) on distal branches of dendritic trees (Rall and Segev, 1987); see also (Segev and Rall, 1988) and (Shepherd et al, 1985). Examples were provided for various kinds of synaptic interactions, supporting a concept of possible logical processing in the distal dendrites of a neuron.

## Motoneuron population models

An entirely different mathematical model was used by Rall in his 1953 Ph.D. thesis. This involved a comparison of theoretical predictions with the input-output relation for the mono-synaptic reflex in a motoneuron population in cat spinal cord. This input-output relation had been demonstrated and discussed by Lloyd (1943, 1945), for the spinal segmental level.

Two probabilistic models (differing in the definition of the threshold) were developed and tested. The simplest assumed that a motoneuron discharges when the number of simultaneously activated synapses on this motoneuron reaches or exceeds a specified threshold number. The other model assumed that a smaller number of synapses could succeed, provided that they were concentrated in a subregion (zone) of the motoneuron's receptive surface. It was found that the experimental data could be fitted by both models. An important test was to fit a set of four input-output curves obtained, from a single preparation, at four different levels of reflex excitability; only one parameter of the theoretical model (threshold value) needed to be adjusted to fit the four curves.

More detail can be found in the original publications (Rall, 1955a, 1955b); also, a useful summary appeared later in a book chapter (Rall, 1990), and as Appendix A.2 in another book (Segev, Rinzel and Shepherd, 1995), where Appendix A.1, by Julian Jack, is also relevant.

In collaboration with Cuy Hunt at the Rockefeller Institute, it was found possible to fit data on the "firing index" distribution in a motoneuron population. It sufficed to assume a normal probability distribution of firing thresholds (Rall and Hunt, 1956).

Note that dendrites had not been distinguished from the motoneuron soma in these early theoretical models; modeling of dendritic cable properties began in 1957; see next section.

## Dendrites as the key to understanding soma voltage transients

In a note to Science (Rall, 1957), it was pointed out that the rapid voltage transient recorded in response to an applied current step from a single cat spinal motoneuron, by means of the recently introduced glass micro-electrode, was being misinterpreted, because the cable properties of the dendrites had been neglected. By assuming this transient to be a single exponential, Eccles and others implicitly assumed that they were recording from a “soma without dendrites”. If , on the other hand, one assumes the dendrites to be dominant, one expects a significantly different transient function, that is much closer to those known to cable theory for non-myelinated axons. The actual problem is a soma with significant dendrites.

Rall prepared a detailed analysis of this intermediate problem, and submitted it to the Journal of General Physiology, in 1958. A negative referee persuaded the editors to reject this MS. The fact that Eccles was this referee was obvious from the many marginal notes found on the returned MS. Fortunately, K. Frank and W. Windle, editors of a new journal, Experimental Neurology, encouraged Rall to expand this MS into two papers: one paper (Rall, 1959) solved the steady state problem for general dendritic branching, and included estimation of membrane resistivity from experimental data; the other paper (Rall, 1960) solved the transient problem, using Laplace transforms, and included estimation of the membrane time constant from experimental data.

It was necessary to work with poorly matched data: electrophysiological data from several sets of impaled motoneurons together with anatomical measurements from different sets of histological motoneurons, in order to estimate values for the underlying model parameters. Assuming uniform passive membrane over dendrites and soma, and assuming that the larger input resistance values correspond to the smaller motoneuron dimensions, it was possible to estimate a range of values from 1000 to 8000 \(\Omega\)-cm^{2}, with a mean of around 5000, for the membrane resistivity; this was significantly greater than values, of 400 to 600, estimated by Eccles and his collaborators for their "standard motoneuron". Also, their "standard" model provided a value of 2.3 for the ratio of combined dendritic input conductance to the somatic input conductance, whereas Rall estimated an extreme range from 10 to 47, with a mid-range from 21 to 35, for this important ratio (Rall, 1959).

It is now clear that the "standard motoneuron" significantly under-estimated the importance of the dendrites. Note that estimating a membrane resistivity that was ten times too small, Eccles obtained values for the dendritic length constant that were 3.2 times too small, leading to exaggerated values for L; for example, a dendritic tree with an actual L=2, would be misunderstood to have 6.4 as its value of L. Such errors contributed to assertions by Eccles that distal dendritic synapses could be dismissed as virtually ineffective. In Rall's modeling, the electrotonic length of the dendritic tree has been in the range from L=1 to L=2, and distal dendritic synapses have been shown to be effective (Rall et al., 1967). This range of values for cat spinal motoneurons has been confirmed by several studies that compared anatomically based estimates with independent estimates based on Eq.(5) above. Note also, that later considerations of synaptic amplification by excitable dendritic spines, and by distal clusters of such spines (section above titled: 8 Dendritic spine model and clusters of spines on branches) add to the efficacy of synaptic input to distal dendritic branches.

The transient analysis (Rall, 1960) provided the basis for a new semi-log plot to estimate the membrane time constant. Instead of plotting the log of the slope, dV/dt. against t (as would be correct for a single exponential transient), one must plot the log of the product: (dV/dt times the square root of t), against t. Examples were provided and discussed. Also, because these analytical solutions were obtained for the case of very long dendrites, it should be noted that the early part of this transient, at the soma, is essentially the same for a cylinder of length, L. This can be understood by considering a long cylinder with equal inputs at X=0 and X=2L. Then symmetry causes the slope, dV/dX, to be zero at X=L, and the early solution at X=0 is only slightly effected by the contribution (spread back) from the distant input at X=2L.

This method of semi-log plotting was applied to some of the transients published by Eccles and his collaborators; larger values for the membrane time constant were obtained, and their implications were discussed (Rall, 1960). Suffice it to say that the special hypotheses, promoted for several years by Eccles, about two phases of synaptic current, were eventually abandoned. A useful review of these issues is provided by Redman and Jack, on pages 27-33, in a book edited by Segev, Rinzel and Shepherd (1995), which also contains reprints of (Rall, 1957, 1959 and 1960).

## Extracellular transients generated by a soma with passive dendrites

At the first International Biophysics Congress, held in Stockholm in 1961, Rall (1962b) presented the results of early computations with an IBM 650 mainframe computer. Assuming an action potential in the soma membrane, and passive electrotonic spread into passive (dendrite) cylinders, the resulting field of extracellular potential was computed, for the case of one cylinder, and then for the case of seven cylinders, for the moment of peak action potential. It was found that the equipotential contours near the soma are almost spherical. Then, with the simplifying assumption of complete radial symmetry, the extracellular transient was computed (at several radial locations), during the timecourse of the action potential at the soma. The resulting extracellular voltage transient was found to be diphasic (-,+), in agreement with the experimental observations of Nelson and Frank (1964) near motoneurons of cat spinal cord.

It was significant that this Rall model generated the kind of (-,+) diphasic transient that others had claimed was evidence for the propagation of action potentials in dendrites. But in this computation, the dendrites were explicitly passive. This meant that such a diphasic record could not be claimed to provide evidence for impulse propagation in dendrites. A discussion of this issue can be found in Nelson and Frank (1964).

The physical intuitive explanation of this computed result is the following: the negative extracellular peak is generated by extracellular current flowing radially toward the soma (during sodium ion current inward across the soma membrane), while the positive extracellular peak is generated by extracellular current flowing radially away from the soma (during potassium ion current outward across the soma membrane, which very rapidly repolarizes the soma, relative to passive continued depolarization of dendritic membrane); see also pages 577& 578 in (Rall, 2006).

## Swartz Prize, 2008

At the Annual Meeting of the Society for Neuroscience, held in November of 2008, the inaugural Swartz Prize for Theoretical and Computational Neuroscience was awarded to Wilfrid Rall (now age 86).

## References

Some of the referenced publications can be found reprinted in a book edited by Segev, Rinzel, and Shepherd (1995). An autobiographical chapter (Rall, 2006) has been published recently by the Society for Neuroscience.

- Coombs J.S., Eccles J.C., and P. Fatt. (1955) The electrical properties of the motoneuron membrane. J.Physiol. 130: 291-325.

- Fatt P., and B. Katz. (1951) An analysis of the end-plate potential recorded with an intracellular electrode. J. Physiol. 115: 320-370.

- Fatt P., and B. Katz. (1953) The effect of inhibitory nerve impulses on a crustacean muscle fibre. J. Physiol. 121: 374-389.

- Goldstein S.S., and W. Rall. (1974) Changes of action potential shape and velocity for changing core conductor geometry. Biophys. J. 14: 731-757.

- Hodgkin A.L., and B. Katz. (1949) The effect of sodium ions on the electrical activity of a crustacean nerve fibre. J. Physiol. 108: 37-77.

- Hodgkin A.L., and A.F. Huxley. (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117: 500-544.

- Holmes W.R., and W. Rall. (1992) Electrotonic length estimates in neurons with dendritic tapering or somatic shunt. J. Neurophysiol. 68: 1421-1437.

- Iansek R., and S.J. Redman. (1973) An analysis of the cable properties of spinal motoneurones using a brief intracellular current pulse. J. Physiol. 234: 613-636.

- Jack J.J.B, Miller S., Porter R., and S.J. Redman. (1971) The time course of minimal excitatory post-synaptic potentials evoked in spinal motoneurones by group Ia afferent fibres. J. Physiol. 215: 353-380.

- Lloyd D.P.C. (1943) Reflex action in relation to pattern and peripheral source of afferent stimulation. J. Neurophysiol. 6: 111-120.

- Lloyd D.P.C. (1945) On the relation between discharge zone and subliminal fringe in a motoneuron pool supplied by a homogeneous presynaptic pathway. Yale J. Biol. Med. 18: 117-121.

- Mendell L.M., and E. Henneman. (1971) Terminals of single 1a fibers: Location density and distribution within a pool of 300 homonymous motoneurons. J. Neurophysiol. 34: 171-187.

- Miller J.P., Rall W., and J. Rinzel. (1985) Synaptic amplification by active membrane in dendritic spines. Brain Res. 325: 325-330.

- Nelson P.G., and K. Frank. (1964) Extracellular potential fields of single spinal motoneurons. J. Neurophysiol. 27: 913-927.

- Phillips C.G., Powell T.P.S., and G.M. Shepherd. (1963) Responses of mitral cells to stimulation of lateral olfactory tract in the rabbit. J. Physiol. 168: 65-88.

- Rall W. (1955a) A statistical theory of monosynaptic input-output relations. J. Cell. Comp. Physiol. 46: 373-411.

- Rall W. (1955b) Experimental monosynaptic input-output relations in the mammalian spinal cord. J. Cell. Comp. Physiol. 46: 413-437.

- Rall W., and C.C. Hunt (1956) Analysis of reflex variability in terms of partially correlated excitability fluctuation in a population of motoneurons. J. Gen. Physiol. 39: 397-422.

- Rall W. (1957) Membrane time constant of motoneurons. Science 126: 454.

- Rall W. (1959) Branching dendritic trees and motoneuron membrane resistivity. Exp. Neurol. 1: 491-527.

- Rall W. (1960) Membrane potential transients and membrane time constant of motoneurons. Exp. Neurol. 2: 503-532.

- Rall W. (1962a) Theory of physiological properties of dendrites. Ann. N.Y. Acad. Sci. 96: 1071-1092.

- Rall W. (1962b) Electrophysiology of a dendritic neuron model. Biophys. J. 2:145-167.

- Rall W. (1964) Theoretical significance of dendritic trees for neuronal input-output relations. In Neural Theory and Modeling, ed. R.F. Reiss. Stanford Univ. Press.

- Rall W., Shepherd G.M., Reese T.S., and M.W. Brightman. (1966) Dendro-dendritic synaptic pathway for inhibition in the olfactory bulb. Exptl. Neurol. 14:44-56.

- Rall W. (1967) Distinguishing theoretical synaptic potentials computed for different soma-dendritic distributions of synaptic input. J. Neurophysiol. 30:1138-1168.

- Rall W., Burke R.E., Smith T.R., Nelson P.G., and K. Frank. (1967) Dendritic location of synapses and possible mechanisms for the monosynaptic EPSP in motoneurons. J. Neurophysiol. 30: 1169-1193.

- Rall W., and G.M. Shepherd. (1968) Theoretical reconstruction of field potentials and dendro-dendritic synaptic interactions in olfactory bulb. J. Neurophysiol. 31: 884-915.

- Rall W. (1969) Time constants and electrotonic length of membrane cylinders and neurons. Biophys. J. 9: 1483-1508.

- Rall W., and J. Rinzel. (1971) Dendritic spine function and synaptic attenuation calculations. Program and Abstracts Soc . Neurosci. First Annual Mtg. p.64.

- Rall W., and J. Rinzel. (1973) Branch input resistance and steady attenuation for input to one branch of a dendritic neuron model. Biophys. J. 13: 648-688.

- Rall W. (1977) Core conductor theory and cable properties of neurons. In Kandel, E.R., J.M. Brookhardt, and V.M. Mountcastle eds. Handbook of physiology, cellular biology of neurons. Bethesda, MD: American Physiological Society; 39-97.

- Rall W., and I. Segev. (1987) Functional possibilities for synapses on dendrites and dendritic spines. In Synaptic Function, ed. G.M. Edleman, W.E. Gall, and W.M. Cowan. New York: Wiley; 605-636.

- Rall W. (1990) Perspectives on neuron modeling. In Binder M.D. and L.M. Mendell, eds. The segmental motor system. New York: Oxford University Press.

- Rall W., Burke R.E., Holmes W.R., Jack J.J.B., Redman S.J., and I. Segev. (1992) Matching dendritic neuron models to experimental data. Physiol. Rev. 72: S159-S186.

- Rall W. (2006) Chapter in "The History of Neuroscience in Autobiography Vol. 5". ed. L.R. Squire, Society for Neuroscience; Elsevier Academic Press.

- Redman S.J., and B. Walmsley. (1983) The time course of synaptic potentials evoked in cat spinal motoneurones at identified group Ia synapses. J. Physiol. 343: 117-133.

- Rinzel J., and W. Rall. (1974) Transient response in a dendritic neuron model for current injected at one branch. Biophys. J. 14: 759-790.

- Segev I., and W. Rall. (1988) Computational study of an excitable dendritic spine. J. Neurophysiol. 60: 499-523.

- Segev I., Rinzel J., and G.M. Shepherd. (1995) The Theoretical Foundation of Dendritic Function. Selected papers of Wilfrid Rall, with commentaries. M.I.T. Press

- Shepherd G.M., Brayton R.K., Miller J.P., Segev I., Rinzel J., and W. Rall. (1985) Signal enhancement in distal cortical dendrites by means of interaction between active dendritic spines. Proc. Nat. Acad. Sci. 82; 2192-2195.

**Internal references**

- Ian Gladwell (2008) Boundary value problem. Scholarpedia, 3(1):2853.

- Frances K. Skinner (2006) Conductance-based models. Scholarpedia, 1(11):1408.

- Peter Jonas and Gyorgy Buzsaki (2007) Neural inhibition. Scholarpedia, 2(9):3286.

- Rodolfo Llinas (2008) Neuron. Scholarpedia, 3(8):1490.

- Ernst Niebur (2008) Neuronal cable theory. Scholarpedia, 3(5):2674.

- Ted Carnevale (2007) Neuron simulation environment. Scholarpedia, 2(6):1378.

- Andrei D. Polyanin, William E. Schiesser, Alexei I. Zhurov (2008) Partial differential equation. Scholarpedia, 3(10):4605.

- Robert E. Burke (2008) Spinal cord. Scholarpedia, 3(4):1925.

- Arkady Pikovsky and Michael Rosenblum (2007) Synchronization. Scholarpedia, 2(12):1459.

## See also

Conductance-based models, Dendritic processing , Electrophysiology, Hodgkin-Huxley model, Neuron, Spines, Synaptic transmision, Synapse