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Rall model

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   Wilfrid Rall (2009), Scholarpedia, 4(4):1369. [3]doi:10.4249/scholarpedia.1369
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   Curator: [7]Wilfrid Rall
   Contributors:


   0.40 -

   [8]Frances K. Skinner
   0.20 -

   [9]Nick Orbeck

   [10]Tobias Denninger

   [11]Eugene M. Izhikevich
     * [12]Dr. Wilfrid Rall, Scientist Emeritus of NIH; Present Address: RR-1, Box
       689, Roseland, VA, 22967

   Figure 1: Diagram of a symmetrically branched [13]dendritic tree. Mapping to an
   equivalent cylinder, and to a chain of ten equal compartments, is indicated
   schematically. Equivalent electric circuit, across the nerve membrane, is shown
   at upper right. (Encyclopedia of [14]Neuroscience, reproduced with permissions).

   The designation, Rall model, usually refers to one or more of several closely
   related biophysical-mathematical models of [15]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 [16]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.

Contents

     * [17]1 Overview
     * [18]2 d^3/2 Constraint on branch diameters
     * [19]3 Equivalent cylinder model and cable equation
     * [20]4 Compartmental model
     * [21]5 Mitral and granule cell dendrites: olfactory bulb model
     * [22]6 Dendro-dendritic synapses predicted and found
     * [23]7 Solutions for input to a single branch of an idealized dendritic tree
     * [24]8 Dendritic spine model and clusters of spines on branches
     * [25]9 Motoneuron population models
     * [26]10 Dendrites as the key to understanding soma voltage transients
     * [27]11 Extracellular transients generated by a soma with passive dendrites
     * [28]12 Swartz Prize, 2008
     * [29]13 References
     * [30]14 See also

Overview

   A conceptual overview of these models can be provided by referring to Figure
   [31]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.([32]1), below). Also, the branching in Figure [33]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 [34]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.([35]2) and Figure [36]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
   [37]1. Here, the [38]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 [39]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: [40]7 Solutions for input to a single branch of an
   idealized dendritic tree, below.

   The circuit diagram, at the upper right of Figure [41]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: [42]5 Mitral and granule cell dendrites: olfactory
   bulb model, and [43]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: [44]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 [45]membrane potential
   along the cylinder must obey a [46]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.
   Figure 2: Theoretical voltage transients computed for analytical solutions based
   on Eq.([47]2), for a tree (or equivalent cylinder) with L=1. Both sets of curves
   show the time-course of membrane depolarization at three locations X=0, X=0.5, &
   X=1. The upper three curves (A) show the computed response to a step increase in
   synaptic excitatory conductance, (from zero to twice the resting conductance),
   applied to only the distal half of the cylinder. The inset (B) shows the computed
   result when the same conductance step is turned off at T=0.2; the upper sharper
   curve is at X=1, while the lower rounded curve is at X=0 (Rall 1962). Note that
   for this square synaptic conductance transient, symmetry implies the sharper
   shape would represent a theoretical EPSP (at the soma) for synaptic input to the
   proximal half of the tree or cylinder, while the delayed and rounded shape
   represents a theoretical EPSP (at the soma) for synaptic input to the distal half
   of the tree. (Reproduced with permissions from Rall (1962)).

   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 [48]separation of variables, a
   general solution of this [49]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 [50]eigenvalues of the [51]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 [52]synaptic excitatory input turned on for a
   short time. Once this synaptic input was turned off, the whole cylinder obeys
   Eq.([53]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 [54]2.

   It is noteworthy that when the expression ([55]3) is introduced into the
   theoretical expression for the exponent in Eq.([56]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.([57]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
   [58]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 [59]NEURON, developed by Hines, and
   [60]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 [61]3, for square
   (on-off) synaptic conductances.
   Figure 3: EPSP shapes computed with the ten-compartment model. Here each
   compartment corresponded to 0.2 units of electrotonic length; with the soma as
   compartment-1, this implies approximately L=1.8 for the dendritic tree. Each of
   these EPSP shapes (at the soma), resulted from a square pulse of excitatory
   conductance (equal to the resting conductance, with a duration of 0.25T)
   delivered only to two adjacent compartments, as indicated in the figure
   (reproduced with permissions, Rall, 1964).
   Figure 4: Two EPSP shape-index loci, at left, based on theoretical EPSP shapes
   like those at right. Here, the synaptic conductance was not square, but a brief
   transient function, F(T) = (T/ Tp) exp (1 - T/ Tp) where Tp equals the time of
   the peak value. For a peak time, Tp = 0.04 of the membrane time constant, this
   transient is shown at right as the dotted curve in mid-diagram. Lower right shows
   three computed EPSPs, each for an input to a single compartment, indicated by the
   number in the triangle; note that the peak amplitudes were normalized to aid
   shape comparison. Two shape index values: "time to peak" and "half width" were
   measured and then plotted in a two-dimensional plot (at left); each point in this
   plot represents a different EPSP shape. The dotted line is a shape-index locus,
   for single compartment inputs, using the same input transient. The solid line
   represents a contrasting shape index locus obtained when the synaptic input was
   delivered uniformly over the entire neuron surface; only the time course of the
   input transient was changed. (Reproduced with permissions from Rall et al. (1967)
   and Rall (1977)).

   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 [62]4.
   Figure 5: Computed contrast between opposite spatiotemporal sequences of
   excitatory synaptic input. Input sequence, ABCD, (at upper left) produced the
   earliest transient time-course, while input sequence, DCBA, (at upper right)
   produced the delayed transient with the higher peak. The component inputs are the
   same as in Figure [63]3; the successive time intervals were one quarter of the
   membrane time constant, as indicated in the figure. The dotted curve shows the
   control transient, where spatiotemporal pattern is eliminated by setting the
   synaptic conductance amplitude 1/4 as large, over all four locations, for the
   full four time periods. (Reproduced with permission, Rall (1964)).

   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 [64]motoneurons of cat [65]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 [66]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 [67]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
   [68]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 [69]extracellular
   (field) potentials that had been recorded in experiments with the [70]olfactory
   bulb of rabbit by (Phillips, Powell & Shepherd, 1963). This involved
   [71]synchronous antidromic activation of a very large population of [72]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 [73]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
   [74]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: [75]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
   [76]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 [77]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.([78]5) above. Note also, that later considerations of
   synaptic amplification by excitable dendritic spines, and by distal clusters of
   such spines (section above titled: [79]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 [80]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. [81]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 [82]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) [83]Boundary value problem. [84]Scholarpedia, 3(1):2853.

     * Frances K. Skinner (2006) [85]Conductance-based models. Scholarpedia,
       1(11):1408.

     * Peter Jonas and Gyorgy Buzsaki (2007) [86]Neural inhibition. Scholarpedia,
       2(9):3286.

     * Rodolfo Llinas (2008) [87]Neuron. Scholarpedia, 3(8):1490.

     * Ernst Niebur (2008) [88]Neuronal cable theory. Scholarpedia, 3(5):2674.

     * Ted Carnevale (2007) [89]Neuron simulation environment. Scholarpedia,
       2(6):1378.

     * Andrei D. Polyanin, William E. Schiesser, Alexei I. Zhurov (2008) [90]Partial
       differential equation. Scholarpedia, 3(10):4605.

     * Robert E. Burke (2008) [91]Spinal cord. Scholarpedia, 3(4):1925.

     * Arkady Pikovsky and Michael Rosenblum (2007) [92]Synchronization.
       Scholarpedia, 2(12):1459.

See also

   [93]Conductance-based models, [94]Dendritic processing , [95]Electrophysiology,
   [96]Hodgkin-Huxley model, [97]Neuron, [98]Spines, [99]Synaptic transmision,
   [100]Synapse
   Sponsored by: [101]Dr. Wilfrid Rall, Scientist Emeritus of NIH; Present Address:
   RR-1, Box 689, Roseland, VA, 22967
   Sponsored by: [102]Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the
   peer-reviewed open-access encyclopedia
   Sponsored by: [103]Frances K. Skinner, Toronto Western Research Institute,
   University Health Network, and University of Toronto
   [104]Reviewed by: [105]Anonymous
   Accepted on: [106]2009-04-28 16:06:42 GMT
   Retrieved from
   "[107]http://www.scholarpedia.org/w/index.php?title=Rall_model&oldid=91692"
   [108]Categories:
     * [109]Computational neuroscience
     * [110]Models of neurons
     * [111]Eponymous

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