Comparison of Alternative Designs for Reducing Complex Neurons to Equivalent Cables

Transcription

1 Journal of Computational Neuroscience 9, 31 47, 2000 c 2000 Kluwer Academic Publishers. Manufactured in The Netherlands. Comparison of Alternative Designs for Reducing Complex Neurons to Equivalent Cables R.E. BURKE Laboratory of Neural Control, National Institute of Neurological Disorders and Stroke, National Institutes of Health, Bethesda, MD reburke@helix.nih.gov Received December 8, 1998; Revised June 21, 1999; Accepted June 29, 1999 Action Editor: Charles Wilson Abstract. Reduction of the morphological complexity of actual neurons into accurate, computationally efficient surrogate models is an important problem in computational neuroscience. The present work explores the use of two morphoelectrotonic transformations, somatofugal voltage attenuation (AT cables) and signal propagation delay (DL cables), as bases for construction of electrotonically equivalent cable models of neurons. In theory, the AT and DL cables should provide more accurate lumping of membrane regions that have the same transmembrane potential than the familiar equivalent cables that are based only on somatofugal electrotonic distance (LM cables). In practice, AT and DL cables indeed provided more accurate simulations of the somatic transient responses produced by fully branched neuron models than LM cables. This was the case in the presence of a somatic shunt as well as when membrane resistivity was uniform. Keywords: electrotonic models, voltage transients, attenuation, voltage propagation delay Introduction Wilfrid Rall (Rall, 1959, 1964) introduced the idea of using an equivalent cylinder model to collapse an idealized branching dendritic tree into a single constant diameter membrane cylinder that has the same total membrane area and electrotonic length (see also Rall and Rinzel, 1973; Rinzel and Rall, 1974). Electrotonic distance X from the soma was used as the metric to accomplish this morphoelectrotonic transformation. Such idealized models have led to important insights about the electrical functions of neuronal dendrites that now thoroughly permeate neuroscience (see commentaries in Segev et al., 1995). In order to collapse a branched tree accurately into an equivalent cylinder, the following criteria must be met: (1) the electrotonic lengths of all paths from the soma to dendritic tips must be the identical; (2) the boundary conditions at each path termination must be the same; (3) the ratio between the diameter of the parent branch at each branch point, when raised to the 3/2 power, must equal the sum of the daughter branch diameters, each raised to the 3/2 power (the so-called 3/2 power rule); and (4) the specific membrane resistance R m and capacitance C m, as well as the specific axial internal resistivity R i, must be uniform throughout the structure. The morphology of actual neurons suggests that all of these criteria cannot be true simultaneously (e.g., Clements and Redman, 1989; Fleshman et al., 1988). With increasing information about the morphology and membrane properties of neurons and the ready access to powerful computer resources, approaches to provide accurate neuron models have become an important issue in computational neuroscience (Koch and

2 32 Burke Segev, 1998; Segev, 1992). Clements (Clements, 1986; Clements and Redman, 1989) introduced a method to collapse an arbitrary (i.e., nonideal) dendritic tree into an unbranched electrotonic cable, referred to as an equivalent dendrite, in which the cable compartments can have unequal diameters. The method is analogous to that used by Rall for the equivalent cylinder; it ensures that cable compartments at each increment of electrotonic distance X have the same surface area as all parts of the original tree at that same increment of X. Clements and Redman (1989) used these computationally efficient cables as surrogates for fully branched cells in trial-and-error estimations of specific membrane and cytoplasmic properties (see also Burke et al., 1994). In constructing equivalent dendrites, Clements and Redman (1989, p. 66) assumed that all points on the dendritic tree at a given electrotonic distance from the soma will be at the same potential at all times (ignoring end effects from dendrites terminating at different electrical distances from the soma, and reflection terms originating at branch points where the 3/2 power law is not followed). However, the ignored effects can produce rather large deviations from this assumption during voltage perturbations (Agmon-Snir and Segev, 1993; Zador et al., 1995). Although variable diameter equivalent dendrites are more accurate than constant diameter equivalent cylinders as surrogates for real neurons, they do not mimic all of the electrotonic properties of fully branched trees (Clements and Redman, 1989; Rall et al., 1992), precisely because they do not include the these effects. The present work was undertaken to test whether other morphoelectrotonic transforms, specifically somatofugal voltage attenuation and signal propagation delay, can be used to collapse complex dendritic trees into an unbranched equivalent cable. These transforms specifically include the end effects produced by finite dendritic paths on voltage distribution and provide heuristically useful visual impressions of voltage distributions in neurons (Zador et al., 1995). The present article demonstrates that it is possible to construct equivalent cables based on these alternative electrotonic metrics and tests how well they mimic the input conductance and transient responses of fully branched cat motoneurons with passive membrane. The results indicate that the new cables outperform cables based on simple electrotonic distance in these respects. Some of this material has appeared in abstract form (Burke, 1997). Methods Basic Cable Attributes: Lamda Cables This work deals with constructing compartmental models of reconstructed (i.e., digitized) neurons in which the data is inherently discretized. In the compartmental equivalent cylinder model of Rall (1959), an ideal branched tree can be collapsed into an unbranched sequence of identical compartments such that the surface area is distributed with respect to electrotonic distance, X, in the same way as in the original tree. The resulting equivalent cylinder has a constant diameter, the same total surface area as the original tree, and the same electrotonic length L as all of the terminating paths in the original tree. Values for specific passive membrane resistance R m and cytoplasmic resistance R i are usually chosen before cable construction because the cable metric depends on the electrotonic lengths λ i of each of the i segments in the tree: R m λ i = di, (1) 4R i where d i is the segment diameter. The electrotonic length X i of the ith segment is simply its physical length divided by λ i. As detailed by Clements and Redman (1989), the same approach can be used to construct an equivalent cable with unequal diameters from an arbitrary tree where the diameter of the jth compartment, D eq (X j,), at electrotonic distance X j, is the 2/3 root of the sum of the 3/2 power of the diameters d i (X j )oftheicylindrical segments in all j dendritic paths in the branched tree within the X bin ending at that X j : [ ] n 2/3. D eq (X j ) = d i (X j ) 3/2 (2) i=1 For the specified X bin size, the length of the cable compartment l eq is R m l eq (X j ) = X λ j = X Deq (X j ). (3) 4R i This method of cable construction ensures that the branched tree and its cable representation have the same total area, and the distribution of surface area

3 Alternative Equivalent Cables 33 Figure 1. Cable construction methods applied to an idealized branching tree with stem diameter = 10 µm, five orders of branching and daughter branch diameters conforming to the 3/2 power rule, and all electrotonic paths of equal length. The electrotonic distance scale was calculated with R m = 10,000 cm 2 and R i = 70 cm. Collapse of the tree structure to a single series of compartments with equal X requires only evaluation of the outward distribution of membrane in terms of electrotonic length of each branch, X i, j. However, calculation of outward attenuation first requires tabulation of the conductance ratio, B out,i,j, starting at the distal end of each branch. Similarly, calculation of outward propagation delay PD i, j requires inward tabulation of the output delay D out,i,j, for each branch, as well as B out,i,j. as a function of X j, S(X j ), is also the same, since R m S(X j ) = πl(x j )D eq (X j ) = π X D eq (X j ) R i (4) Conservation of surface area is an important feature of the equivalent cylinder representation. These cables will be referred to as lamda cables (LM cables) because they are based on the electrotonic distance metric. In practice, the initial step in constructing lamda cables is to tabulate the somatofugal electrotonic distance X j of all j cylindrical segments of the digitized neuron dendritic tree and to determine the points of intersection with the selected X bins (Fig. 1A). In a constant diameter segment, X j varies linearly with physical length l j and thus can be used to subdivide the segment along its length. The 3/2 power of the diameter of those portions of each segment within a given X j bin is summed into an array representing somatofugal X. When complete, this array is used to calculate the diameters and physical lengths of the final LM cable, using Eqs. (2) and (3), as described by Clements and Redman (1989; see also Burke et al., 1994). The goal in the present study was to construct analogous equivalent cables based on two alternative electrotonic metrics: (1) attenuation (AT) cables based on the outward attenuation of voltage propagating into the dendritic tree from the soma and (2) delay (DL) cables based on the time delay of somatofugal signal propagation into the dendritic tree. The possibility of using the attenuation and signal delay metrics was suggested by the utility of these morphoelectrotonic transforms to display neuronal morphologies so as to emphasize the effects of dendritic structure on signal transfer characteristics (Agmon-Snir, 1995; Zador et al., 1995). As with LM cables, both AT and DL cables are constructed so as to conserve surface area. Dimensionless Conductance Ratios The construction of AT and DL equivalent cables is less straightforward than for lamda cables because four stages are required. Unlike LM cables, both AT and DL cables require information about the boundary conditions at the distal ends of each dendritic segment in order to tabulate the respective metrics. The distal end is defined by the direction toward which current

4 34 Burke flows within the segment. In the case of current flowing from the soma into the dendrites, the process of cable construction begins at the terminations of each dendritic path (Figs. 1B and 1C). Both AT and DL cables require the value of the dimensionless conductance ratio B out at the distal end of each cylindrical compartment (Rall, 1977): B out = G out G, (5) where G out is the conductance for current flowing out of its distal end and G is the conductance of a semiinfinite extension of a cylinder with the same diameter d: G = πd3/2 2 R m R i. (6) The initial value of B out is defined for terminating segments by the assumed boundary condition, which in the present work was sealed end (B out = 0). Calculation of the dimensionless input conductance B in,i at the other end of each cylindrical segment B in,i = B out,i + tanh(l i ) 1 + B out,i tanh(l i ), (7) where L i is the electrotonic length of that segment. The B out,i 1 of the next more proximal cylinder i 1 is then [ ] 3/2 di B outi 1 = B ini, (8) d i 1 where d i i and d i are the diameters of the respective cylinders. The process is iterated toward the soma until a branch point is encountered, whereupon the iteration begins at the termination of the other daughter branch of that branch point. The dimensionless output conductance of the parent branch B out,par depends on the input conductances and diameter ratios of its two daughter branches: B out,par = B in,dau1 [ ddau 1 d par ] 3/2 + B in,dau 2 [ ddau 2 d par ] 3/2. (9) In the present work, the B out,j values were tabulated for all segments of the digitized dendritic tree. These conductance ratios, introduced by Rall (Rall, 1959, 1977), can be used to calculate the steady-state input conductance at any point in a branched tree, including trees with arbitrary branch characteristics and nonuniform membrane properties (Fleshman et al., 1988). The values of B in for the stem segments of all trees belonging to the neuron are summed to give the input conductance into the entire dendritic tree. This sum is added to the soma conductance to give the total input conductance of the neuron (see Fleshman et al., 1988). Attenuation Cables The steady-state voltage attenuation atten in a membrane cylinder (i.e., having constant diameter) depends on its electrotonic length L and the boundary condition, B out, at the end distal to the direction of current flow (Rall, 1959, 1977): atten = V in = cosh(l) + B out sinh(l), (10) V out where V in and V out are the steady-state voltages at the proximal and distal ends of the cylinder. Note that atten is always >1.0. The attenuation metric actually used for cable construction was A = ln(atten), as in earlier work (Agmon-Snir, 1995; Zador et al., 1995). With the tabulated B out values for the entire dendritic tree, calculation of outward (somatofugal) A can begin at the soma using Eq. (10) and then progress outward (i.e., in the direction of current flow; see Fig. 1) adding the values along the jth path to its termination, such that n n A j = ln(atten k ) = A k, (11) k=1 k 1 where the index k refers to all n segments on the direct path from the soma up to and including its termination segment. The primary objective was to construct an equivalent cable that conserves the surface area of the original branched tree as in LM cables but distributed according to the outward attenuation A j in the original tree. Unfortunately, the tabulated A j values for segments in the branched tree cannot be used in the same way as X j in constructing LM cables because A j is a nonlinear function of physical segment length. The strategy to overcome this problem was to divide each segment in the tree into small increments of physical length (5 µm was used in the present work) that can each be regarded as piecewise linear with respect to A j. The values of A j and B out,j tabulated for each segment of the tree were

5 Alternative Equivalent Cables 35 used to calculated the location of each increment in the sequence of A bins selected for the cable. The d 3/2 and surface area for the increment were then summed into the appropriate A bin or proportionately to adjoining bins if the fragment crossed a bin boundary. This process was iteratively applied to each of the segments in the branched tree until the final cable array contained the summed d 3/2 and surface areas for the entire dendritic tree, distributed into A bins exactly as found in the original tree(s). The 2/3 root of the summed d 3/2 gave a provisional physical diameter D for each cable compartment (see Eq. (2)). The physical length of the compartment was then calculated from the summed area value S( A n ). The actual ln(atten) ofthe Acompartments in this raw cable array often did not exactly match the desired A. Therefore, a successive approximation procedure was used to adjust the physical lengths and diameters of each compartment in order to produce the exact A, while maintaining the area of each compartment at its original value. Because of the dependence on boundary conditions, this process began with the terminal compartment, where B out = 0. The A of the terminal compartment was calculated from its S and D values, as in the process for the full tree (Fig. 1B). The normalized difference between A j and the desired A diff = A j A (12) A was multiplied by an appropriate weighting factor (usually 0.1, chosen to ensure convergence) in order to adjust the compartment s length and diameter, dividing the length and multiplying the diameter to maintain the area constant. The process was iterated until A j and A differed by ɛ< With the correct dimensions in the terminal cable compartment, the process was iterated for successively more proximal compartments until all had the desired A. Delay Cables The process of calculating the delay cable for a particular structure was the same four-stage process used for attenuation cables, but the metric used was the propagation delay (PD), which is a measure of the time delay (in ms) of propagation of the centroid of any transient voltage within a membrane cylinder (Agmon-Snir, 1995). It is important to note that C m must also be specified because the PD transform depends on the membrane time constant, τ = R m C m, as well as on the output conductances. Like A, PD varies nonlinearly with physical distance along each cylindrical compartment and the solution to this problem was the same as detailed for the AT cable construction. The only difference (other than the equations used) was that the necessary boundary conditions also include the output delay D out, which is the time domain analog of the steady-state output conductance B out. In any given membrane cylinder, PD depends on its electrical length L, the local membrane time constant τ = R m C m, its output conductance (represented by B out ), and its output delay D out (Agmon-Snir, 1995). Some of the equations using in the present work (Eqs. (14) and (15)) were modified from those given by Agmon-Snir (1995, his Eq. (36)) in order to utilize the dimensionless conductance ratios already calculated. Analogous to the AT cables, the first step was to tabulate values for D out and B out for each cylinder in the original branched structure, starting with the terminations of each dendritic path (Fig. 1C). Values of D out were obtained with the following equation (Agmon-Snir, 1995, his Eq. (35)): [ ] 1 D out = τ κ(1 + ξ) ξl ξ 2 exp( 2L) exp(2l) (13) for a terminal compartment with sealed end where B out = 0, κ = 0, and ξ = 1. For each more proximal compartment κ = B [ 1 out 2 ] D out τ, (14) B out + 1 where Dout is the output delay of the immediately distal compartment and ξ = 1 B out 1 + B out. (15) The process was iterated for all paths in the dendritic tree, using Eqs. (7) to (9) as well as those above. Using the tabulated values, calculation of the outward propagation delay PD began at the somatic end of the tree (Fig. 1) using the following equation (Agmon-Snir, 1995, his Eq. (37)): [ [ξ + 1] κ ξ L PD = τ κ + L ]. (16) ξ + exp[2l] 2

6 36 Burke These values were tabulated for each segment in the tree and their sum was accumulated along each path to a dendritic termination. Since PD, like A, is a nonlinear function of physical length along each segment in the tree, the distribution of d( PD n ) 3/2 and S( PDn ) into the PD n bins of the cable array was done using short pieces of the successively more distal segments of the tree in the same way as done for the AT cables. Again, the PD of compartments of the raw cable array usually did not exactly match the target PD desired. Therefore, the same process of successive approximation was used to adjust the lengths and diameters of each cable compartment in order to match the target PD. In the resulting delay cable, each final compartment has the appropriate PD and the same surface area as the original components of the dendritic tree with that somatofugal PD. Additional Attributes The computer program that was written to implement the above procedures also calculated the input conductance into the original dendritic tree(s) of the anatomical neuron data file, using specified values for R m, R i, and C m (1.0 µf/cm 2 in the present work). It also generated two output files, one of which encoded the structure of the desired equivalent cable for import into the neuron simulation package, NODUS (De Schutter, 1992), that was used for testing the transient behavior. The other file contained the dimensions of cable compartments, their actual X, A, and PD values, the cumulative somatofugal values for the same metrics, and the cumulative area and physical length of the cable (e.g., Figs. 2 and 3). It was of particular importance to ensure that the effective surface area of the soma was the same for full and cable models of the same cell. NODUS calculates this by subtracting the cross-sectional areas of all stem dendrites arising from the soma from the surface area implied by the specified soma diameter. This effect of different stem areas in the full cell cable models was taken into account in the program. This source code, written in Pascal for the Macintosh, is available from the author on request. Test Criteria The object of this work was to compare the steadystate input conductance and transient responses of models with fully branched dendrites with models that incorporate their equivalent cables plus a soma of the same dimensions as the full model, all constructed using the same values for R m, C m, and R j. Transient responses to identical brief current pulses at the model soma produced by the equivalent cable models were compared with those generated by the fully branched model, using relatively small values of X ( 0.07) for the latter. Transients were calculated using the Fehlberg (exact) integration method with adaptive time steps. The percentage error between the cable and full model transients, V err (t), was calculated as a function of time by [ ] Vcbl (t) V err (t) = V full (t) 1 100, (17) where V cbl (t) and V full (t) are the transients produced by the cable and full models, respectively (e.g., Figs. 4A and 4C). The difference between the transients responses produced by the full model and the different cable representations at t = 0.1 ms were also evaluated by calculating the RMS error: RMSerr(t i ) = i=max i=min V err(t i ) 2, (18) count where count is the number of values already summed to the time step in question. For any given time epoch, the RMS error gives a single value summation of the disparity between the V (t) curves produced by cable models in relation to the full structure. It was also useful to examine the evolution of the RMS error during the time course of the transients. This was done backwards in time from a point at which the curves had converged to near zero error ( back RMS error ; see Figs. 4B and 4D), which gives a clear impression of the relative fidelity of the different models over the transient time course. For any given time epoch, the value of RMSerr(t) was the same whether it was calculated forward or backward in time. Results Idealized Tree Structure The algorithms for cable construction were first applied to the idealized tree structure shown in Fig. 1, using R m = 10,000 cm 2, R i = 70 cm, and C m = 1.0 µf/cm 2 as parameters. This tree had five orders of branching, each with X = 0.25, with

7 Alternative Equivalent Cables 37 Figure 2. Physical and electrotonic dimensions of three equivalent cables based on the idealized tree shown in Fig. 1, assuming spatially uniform R m = 10,000 cm 2, R i = 70 cm, and C m = 1.0 µf/cm 2. A: Diameters of cable compartments (ordinate) plotted against physical cable length (abscissa). The diameters of the LM, AT, and DL cables were all exactly 10.0 µm except for the terminal compartment of the AT and DL cables (symbols are offset to permit visualization). The X (LM cable), A(AT cable), and PD (DL cable) bin sizes were adjusted (symbol key) to generate cables with the same number (n = 26) of compartments in order to facilitate comparison. B, C, and D: Comparison of electrotonic length X, outward attenuation A (both plotted on left ordinate), and propagation delay PD (right ordinate) of individual compartments in the final equivalent cables for the idealized tree, plotted against electrotonic length (X, abscissae). Note that the characteristics of compartments in the AT and DL cables were similar but not identical. dimensions that allowed it to be collapsed into an equivalent cylinder with total electrotonic length L = 1.25 and constant diameter (d = 10 µm) equal to that of the stem branch. Accordingly, all three equivalent cable types (LM, AT, and DL) had the same overall dimensions, except for terminal compartments in the AT and DL cables that represented small amounts of residual membrane (Fig. 2A). The bin increments for each cable were adjusted to give a total of 25 compartments to facilitate comparison ( X = 0.05, A=0.026, PD = 0.22 ms). The diameter of the terminal compartments of the AT and DL cables were <10 µm because they had been adjusted to give the assigned value of A or PD (see Methods). Although the overall dimensions of the three cables were essentially identical, the lengths and electrotonic properties of their individual compartments were quite different. This is evident in panels Figs. 1B, 1C, and 1D, which show the actual X, A, and D for individual compartments in the LM, AT, and DL cables, respectively, plotted against location along the electrotonic length of the cables. In each cable, the increment for the specified metric (e.g., X for the LM cable in panel B) was constant, but the increments for the other two metrics were not. In a finite electrotonic cable with

8 38 Burke Figure 3. Physical and electrotonic dimensions of LM, AT, and DL cables for an alpha motoneuron, M43c5. Cable bins were adjusted to give the same number of compartments (n = 26), as in Fig. 2A (increments noted in the symbol key). B, C, and D: Individual compartment X, A, and PD, respectively (as in Fig. 2). Note qualitative similarity to the patterns found for the ideal tree cables (Figs. 2B 2D). sealed end boundary condition, voltage attenuation and signal propagation delay both decrease per unit length as the cable termination is approached (Agmon-Snir and Segev, 1993; Zador et al., 1995). The compartment properties in the LM cable shown in Fig. 2B demonstrate that this end effect actually influences the dimensions of compartments along the entire equivalent cable. The same influence produces the steady growth of compartment X in the AT and DL cables (Fig. 2C and 2D). Note that the characteristics of compartment A and PD were similar but not identical in the AT and DL cables. As expected, the steady-state input conductances for the fully branched tree, and the LM, AT, and DL cables were essentially the same (Table 1), as were the transient decays following a brief somatic current pulse (10 na, 0.3 ms duration). For transient simulations, a 10 µm soma was attached to the fully branched ideal tree and its three cable surrogates, using NODUS software (see Methods). For this comparison, the electrotonic length of the individual compartments ( X) of the fully branched ideal tree and the LM cable was 0.01, while A = 0.01 and PD = 0.1 ms were used for the AT and DL cables, respectively. Equivalent Cable Models of Alpha Motoneurons The program that produced the cables in the ideal tree case was used to generate LM, AT, and DL cables for two fully reconstructed alpha motoneurons (M43c5 and M35c4). The morphology (Cullheim et al., 1987a, 1987b) and estimated membrane electrical properties (Fleshman et al., 1988; Segev et al., 1990) of both cells have been published. Motoneuron M43c5 had

9 Alternative Equivalent Cables dendrites, with a total of 163 paths that terminated between X = 0.4 and X = 2.8 (median about 1.4), assuming R m = 11,000 cm 2, R i = 70 cm, and C m = 1.0 µf/cm 2 (Fleshman et al., 1988). Assuming a spherical soma with the average diameter measured for this cell, the ratio of dendritic to somatic membrane area, A dend /A soma (609,807 and 7,481 µm 2, respectively), was The physical dimensions of the three equivalent cables for cell M43c5 (Fig. 3A) illustrate the marked contrast between cables for an actual motoneuron and those for an idealized tree (cf. Fig. 2A). The diameters of each of the surrogate cables were roughly constant only in the proximal one-fourth and then fell more-orless linearly to their terminations (see also Fig. 9 in Fleshman et al., 1988). The AT and DL cables differed from the LM cable mainly in their most distal portions. For purposes of illustration (as in Fig. 2A), the cable bin increments were adjusted to produce cables with 26 compartments in each case. The graphs of cumulative area as a function of cable length were curvilinear and closely similar for all three cables (not shown). The patterns of X, A, and PD for individual compartments in relation to electrotonic location (Figs. 3B 3D) were basically similar to the idealized tree (Figs. 2B 2D), allowing for the irregularities introduced by the variety of electrotonic path lengths in the real neuron. The steady-state input conductance of the entire dendritic tree of this cell was most closely matched by the AT cable surrogate (Table 1; error = 0.04%). Because of the large size of this motoneuron and its relatively low estimated R m, X=0.07 was used to represent the cell for transient simulation. This produced a model cell with 2,664 compartments, which is near the maximum capacity for NODUS. The fully branched dendrites were connected to a spherical soma with diameter of 48.8 µm, in which a 30 na, 0.3 ms pulse was delivered at t = 0. The resulting decay transient was compared with transients generated by the Figure 4. Plots of the disparity between transient responses produced in the full cell model versus in three equivalent cable models of an alpha motoneuron (cell M43c5). Parameters for transient simulations are given in the text and the cable increments used are indicated in the symbol key in B. A: Plots of V (t) obtained from the fully branched cell model (heavier line) and for the three cable surrogates. The largest disparities were evident between 2 and 8 ms, as depicted in the inset. B: V err (t) curves showing the percent error between the transients produced by each cable and that of the fully branched model (see Methods). Note that AT and DL cable transients converged near zero error well before that of the LM cable response but all error curves eventually converged to zero error at about three times the membrane time constant (arrow). C: Backward RMS error curves (RMSerr(t); see Methods) calculated from the end of the transient simulations (50 ms) down to 0.4 ms (backward error) for the responses shown in A. Note that the curves for AT and DL cables exhibited much lower RMS error than that for the LM cable for most of the time courses. The AT cable was slightly more accurate than the DL cable.

10 40 Burke Table 1. models. Comparison of summed input conductance for full tree and equivalent cable Ideal tree M43c5 M35c4 GMN 9118 G in Error G in Error G in Error G in Error Model (ns) (%) (ns) (%) (ns) (%) (ns) (%) Full tree LM cable AT cable DL cable same pulse in each of the three types of equivalent cables connected to a soma with the same effective surface area (see Methods) and the increments given in the symbol key in Fig. 4B. The curves in Fig. 4A illustrate the V (t) decay transients produced by the fully branched cell model and the LM, AT, and DL surrogate models. The inset shows the relatively small differences between the curves within the dashed line box in more detail. It is difficult to evaluate the relative fidelity of the cable models from such a display. However, they are magnified in plots of the instantaneous error, V err (t), between the decay transient from the full cell and those from the three cable models, evaluated from 0.4 to 50 ms (Fig. 4B; see Eq. (17)). The errors were relatively large immediately after the simulated current pulse (t = 0.4 ms), but each curve eventually converged to near zero error by about t = 33 ms (arrow; three times the τ 0 of 11 ms). Thus the longest time constants, τ 0 (in this case τ 0 = τ m = R m C m = 11 ms) were identical in all four structures, but the shorter equalizing time constants that represent the spread of current into the dendrites were clearly not the same. The AT cable appeared to mimic the mixture of equalizing time constants in the full tree better than the DL or LM cables. Because of the disparity in shapes between the V err (t) curves, it proved to be useful to evaluate the relative errors by calculating the RMSerr(t) backwards in time from a point at which all of the error curves had converged to near zero (Eq. (18); see Methods). Figure 4C illustrates backward RMSerr(t) curves for the three cables, calculated from 50 ms down to 0.4 ms. Viewed in this way, the LM curves clearly showed the largest error down to about 2 ms, when the RMSerr(t) for the DL curve became larger. The AT cable had the lowest RMSerr(t) throughout. The total RMS errors accumulated between 2 ms and 33 ms (3 τ 0 ; about Table 2. Simulations with spatially uniform R m. M43c5 M35c4 GMN9118 R m 11,000 20,000 33,000 ρ (uniform R m ) A dend /A soma RMS err (3τ) LM (%) RMS err (3τ) AT (%) RMS err (3τ) DL (%) where the three curves converged) for the three cable models of this cell are given in Table 2. The value of 2 ms was chosen to start this calculation because, in practice, the first 1 to 2 ms of experimental transient records are usually ignored because they are often contaminated by artifacts. Similar results were obtained with another alpha motoneuron, a type S soleus motoneuron with estimated R m = 20,000 cm 2 (cell M35c4; Fleshman et al., 1988). Although this cell had a simpler dendritic tree than M43c5, the transient error comparisons were similar in shape and magnitude to those illustrated in Fig. 4. In this case, the V err (t) curves converged to near zero at about t = 60 ms, again about three times τ m = 20 ms. As with M43c5, the AT cable showed the smallest error during transients as evaluated by backward RMSerr(t) curves as well as by total RMS error (Table 2). The AT cable also best matched the total input conductance of the fully branched dendritic tree (error 0.04%; Table 1). Cable Models of a Gamma Motoneuron The final example is a gamma-motoneuron that has been described in detail anatomically (cell GMN9118; Moschovakis et al., 1991) and electrophysiologically (Burke et al., 1994). This cell had a dendritic tree that

11 Alternative Equivalent Cables 41 was much smaller and simpler than either of the alpha motoneurons discussed above (nine dendrites with a total of 33 terminating paths). The ratio of dendritic to somatic membrane area (105,382 and 2,961 µm 2, respectively) was smaller than in the alpha motoneurons (35.6; Table 2). With an estimated R m of 33,000 cm 2, these paths ended between X = 0.6 and X = 1.4 (as before, R i = 70 cm and C m = 1.0 µf/cm 2 ). The dimensions of the AT and DL cables for this cell showed considerable differences from the LM cable over most of the distal two-thirds (Fig. 5A; cf. Fig. 3A). As in the other examples, the total steady-state input conductance into the dendrites of the AT cell model most closely matched that of the fully branched cell (Table 1). Transients in the full cell were simulated in a model with 2,367 compartments ( X = 0.01) and a spherical soma with diameter = 30.7 µm. The instantaneous errors, V err (t), between the cable transients and the fully branched tree were quite large for the LM in comparison with the AT and DL cables (Fig. 5B) but all three curves converged to near zero error at t 90 ms (approximately three times τ 0 = 33 ms). Evaluation of the backward RMS error from 90 ms (Fig. 5C) showed that the LM model produced the largest cumulative error throughout the duration of the transients, while the AT cable outperformed the DL cable for times >10 ms from the onset. The total RMS error over 2 to 90 ms of the transient was smallest for the DL cable, only slightly higher for the AT, and quite a bit larger for the LM cables (Table 2). Effect of Reducing Cable Compartment Number Figure 5. Comparison of three equivalent cables for a cat gamma motoneuron (cell GMN9118). A: Physical dimensions of the three equivalent cables, as in Figs. 2A and 3A. Cable increments were again adjusted to produce the same number of compartments (n = 26); increments given in the symbol key. B: Verr(t) for 90 ms transient responses using spatially uniform R m = 33,000 cm 2, R i = 70 cm, and C m = 1.0 µf/cm 2 ; cable increments for simulations given in symbol key in C. Note that the error curves converged near zero only after 90 ms (about three times the system time constant of The comparisons discussed above were done using equivalent cables with small increments of the morphoelectrotonic metrics (e.g., X = 0.01), in order to minimize errors due to the assumption of compartment isopotentiality (see Segev et al., 1998). However, one objective of equivalent cable representations is to reduce the complexity of dendritic neurons in order to enhance computational efficiency (Segev, 1992; Douglas and Martin, 1992; Bush and Sejnowski, 1993). Accordingly the effect of increasing the morphoelectrotonic bin size (i.e., using increasingly 33 ms), as in the responses from alpha motoneuron models shown in Fig. 4. C: Backward RMS error (90 ms down to 0.4 ms), showing that the AT and DL cables outperformed the LM cable over most of the transient duration.

12 42 Burke coarse cables) in the three types of cable models was examined. Figure 6 illustrates families of backward RMS error curves for increasingly coarse representations of cell M43c5. In each case, increasing the cable compartment size over an order of magnitude produced relatively small reductions in transient fidelity; serious increases in errors were found only when there was a 20-fold increase in bin size. Paradoxically, the coarsest LM cable exhibited slightly smaller backward errors in comparison with those having smaller values of X. Reducing the number of model compartments markedly decreased computation time for transient simulations. These results demonstrate that relatively coarse equivalent cables, when properly constructed, can provide rather good fidelity with the fully branched model. Effect of a Somatic Shunt Recent evidence suggests that the use of sharp microelectrodes may introduce somatic shunt conductances (Spruston and Johnston, 1992; Staley et al., 1992) that greatly complicate the interpretation of electrotonic behaviors (Durand, 1984; Iansek and Redman, 1973; White et al., 1992). A somatic shunt was simulated in the present work by decreasing the value of R m,soma for transient simulation runs in NODUS (Fleshman et al., 1988). Transient responses produced by the ideal branched tree in Fig. 1A and its cable surrogates (Fig. 2) were examined with R m,soma = 500 cm 2 and R m,dendrite = 10,000 cm 2. All of the cable responses were very closely similar to each other as well as to that of the full tree structure, except that the system time constant τ 0 in all cases was 9.45 ms, rather than 10 ms for the uniform R m case. This orderly behavior, entirely expected for an idealized tree, was not found with real neurons. In order to compare the results in different motoneurons, the ratio of dendritic to somatic local time constants β was maintained constant (see Rall et al., 1992). Figure 7 illustrate V err (t) curves for alpha (panel A) and gamma (panels B and C) motoneurons already presented in Figs. 4 to 6, using the same R m,dendrites as in the previous simulations but with R m,soma adjusted to produce β = 66 (Table 3). In the presence of a somatic shunt, the errors between the transients produced Figure 6. Backward RMS error curves for the three cable types based on cell M43c5, showing the effect of making the cables more coarse by increasing the compartment bin sizes. Increasing bin size over a tenfold range produced little degradation in transient fidelity, but a further increase to 20 times the minimum size showed a large error. Cable increments and the resulting number of compartments in each cable (n) are indicated for each curve.

13 Alternative Equivalent Cables 43 Table 3. Somatic shunt simulations. M43c5 M35c4 GMN9118 R m,dendrites 11,000 20,000 33,000 R m,soma β ρ (shunt) RMS err (3τ) LM (%) RMS err (3τ) AT (%) RMS err (3τ) DL (%) by the fully branched and the cable models were both larger and had more complex time courses than the cases examined with uniform R m. In particular, none of the error curves converged to zero error, and none settled to the same system time constant τ 0 as the full cell model (i.e., none of the curves became parallel to the abscissa; see Figs. 4B and 5B). As was the case with spatially uniform R m, the V err (t) curves of the AT and DL cables resembled one another, but the error trajectories for the LM cables were quite different. The results with the third alpha motoneuron M35c4 were similar to those found for M43c5 (see Table 3). In order to put the percent errors plotted in these curves into perspective, the V (t) transient from the full model of cell GMN9118 is shown in Fig. 7C superimposed on the transient produced by the LM cable model for that cell, both with the same somatic shunt. The two transients cross and recross in several places (arrows), an effect much more obvious in the LM error curve in panel B. Despite the larger errors, the AT cable models outperformed the other two even in the presence of a somatic shunt. Discussion Figure 7. Instantaneous error curves produced by cables with nonuniform R m (adding a somatic shunt). A and B: V err (t) curves for cells M43c5 (A) and GMN9118 (B) when the somatic membrane resistance R m,soma was reduced in each model from the spatially uniform values used in Figs. 4 to 6 to R m,soma = 167 and 500 cm 2, respectively (see Table 3). See text for full discussion. C: V(t) curves from the full cell model of GMN9118 compared to that of the LM cable for the same cell, both with R m,soma = 500 cm 2. This illustrates the largest disparity found in the present work. This article demonstrates that it is possible to collapse arbitrarily complex dendritic trees into equivalent cables based on outward voltage attenuation and on outward signal propagation delay, which to my knowledge have not previously been used as bases for construction of cable surrogates. In addition, the AT and DL equivalent cables outperform LM cables in mimicking the steady-state input conductances and transient responses of fully branched models of actual motoneurons whether or not R m is spatially uniform.

14 44 Burke This greater fidelity presumably results from the fact that the construction of AT and DL cables take into account the end effects of terminating dendritic paths (Agmon-Snir, 1995; Agmon-Snir and Segev, 1993; Zador et al., 1995), which do not enter at all into the construction of LM cables that are based only on somatofugal electrotonic distance (Clements and Redman, 1989). The input conductances of AT cables based on actual motoneurons were virtually the same as those of the full branched structures (Table 1), which suggests that AT cables more accurately mimic the steady-state voltage distribution within the full model than the other cable types. Why Do Errors Appear in Cable Simulations of Neuronal Dendritic Trees? All three types of equivalent cables produced from the ideal dendritic tree (Figs. 1 and 2) showed input conductances (Table 1) and transient responses that were essentially identical to those of the fully branched structure and therefore to each other. This could hardly have been otherwise because all of these cables had the same diameter and total length, except for the terminal compartments that represented small amounts of residual membrane left after segmentation of the branched tree according to AT or DL (Fig. 2A). The ideal case was used to validate the present computational strategy. The situation was different in the case of actual motoneurons with spatially uniform R m. With respect to fidelity of the transient responses, the AT cables were marginally superior to the DL cables over much of the transient durations but both showed some residual error, albeit less than the LM cables (Figs. 4 and 5). As Rall has shown (Rall, 1969), the transients responses at any location x, in an electrotonic cable structure, branched or unbranched, can be represented as the sum of a series of exponentials: n max V (x, t) = V (x, t) C n exp(t/τ n ), (19) n=0 where C n are coefficients, or weights, that control the magnitude of each time constant s (τ n ) contribution to the net response and V (x, t) is the steady-state transmembrane voltage. For a continuous cable n max = while in a compartmental simulation, n max equals the number of compartments and the values of C n and τ n can be obtained from the eigenvalues of the simulation matrix (see Rall et al., 1992). Therefore, models with different numbers of compartments must have different mixtures of C n and τ n, which may or may not summate into similar net responses (e.g., Rall et al., 1992, their Fig. 7). In the case of fully branched motoneurons with spatially uniform R m, they did not (Figs. 4 and 5), although the V err (t) curves for the motoneurons converged near zero error as time increased to about three times τ 0. This indicates that τ 0 was identical in branched and cable models, as expected when the same values of R m and C m apply to all regions of branched and cable models of a given cell. The larger errors at earlier times presumably resulted from differences in the mixture of shorter time constants and/or their coefficients. The fact that the relative fidelity of equivalent cable transients showed little degradation as the number of compartment decreased (up to a point; see Fig. 6) indicates that the number of coefficient time constant pairs per se is not critical. In fact, V err (t) actually decreased slightly in LM cable responses as X increased (Fig. 6A). The errors in transients produced by the present equivalent cable models must result from the inability of these reduced cables (and possibly of any reduced cable model) to accurately reproduce all of the end effects and reflection terms that are inherent in a natural branched dendritic tree. The AT and DL cables, do a better job than the LM cables, but they are not perfect. Recently, an equivalent cable formulation that takes account of all of the electrotonic pathways within a tree structure was introduced (Ogden et al., 1999; Whitehead and Rosenberg, 1993). Such a structure in principle should provide an exact match for the electrical behavior of the tree as observed from any point within it. Preliminary results using simple asymmetrical trees, attached to a small soma and treated exactly as described in this article, showed that transient responses of such Lanczos equivalent cables were virtually identical with those from the branched structure (R.E. Burke and J.M. Ogden, unpublished observations). However, the Lanczos equivalent cable can require many thousands of compartments to represent each current path in the structure, thus providing no structural simplification even for modestly complex real trees. Transients in Models with Simulated Somatic Shunt The introduction of a simulated passive shunt conductance in the soma complicates neuron models because the soma then has a shorter effective local time

15 Alternative Equivalent Cables 45 constant than the dendrites (Durand, 1984; Iansek and Redman, 1973). The slowest time constant τ 0 in the somatic transient response to a short current pulse (sometimes called the system time constant; Fleshman et al., 1988) is somewhere between τ soma = R m,soma C m and τ dendrites = R m,dendrites C m (assuming spatially uniform R m,dendrites ). When R m is the same throughout the neuron, equalizing currents flow out into the dendrites after a somatic voltage perturbation until all of the membrane capacitance is equally charged, after which transmembrane voltage eventually decays everywhere at the same rate, which is the membrane time constant, τ m = R m C m, which occurs at t 3 τ 0 (Figs. 4 and 5). Initial somatofugal equalizing currents also flow during and after a voltage perturbation in a soma with a local shunt conductance. However, because the somatic capacitance discharges more rapidly than the dendritic, equalizing currents soon begin to flow back toward the soma from regions with longer local time constants even as the more distal dendritic regions continue to be charged (Fleshman et al., 1988). The cell interior returns to an isopotential state only when all of the injected charge has completely dissipated. The sensitivity of the V err (t) curves used in the present work reveals the complexity of this process, which is not usually appreciated. Because the soma receives dendritic current during the process of charge dissipation, its potential decays at a rate somewhere between τ soma and τ dendrites. The value of τ 0 in model of an ideal cylinder attached to spherical soma with a shunt conductance depends on the electrotonic length of the cylinder, L, β, and the value of ρ in the absence of a somatic shunt (Holmes and Rall, 1992). In the present ideal tree models with R m,soma = 500 cm 2 and β = 20, τ 0 recorded in the soma was 9.45 ms in the full tree and all its cable surrogates, entirely compatible with the calculations of Holmes and Rall. It was therefore surprising that the V err (t) curves for equivalent cable models of actual motoneurons did not converge to the same τ 0 as the full cell model (Figs. 7A 7C), as found for the uniform R m models (Figs. 4 and 5). It is important to recall that the somatic, dendritic, and total membrane areas of each fully branched cell and its equivalent cables were the same in all of the models used for transient simulation. This proved to be a critical factor in the presence of somatic shunts; small discrepancies in effective somatic area produced larger errors. Given that the respective local time constants were also the same, one would expect that τ 0 would be the same for all models of a given cell, as was found for the ideal tree. In contrast, the sensitivity of the V err (t) curves revealed that the late voltage decays of the cable model responses not only differed from the full tree but also differed from each other. Indeed, the V err (t) curves for each cable model continued to exhibit curvature until numerical truncation errors made it impossible to evaluate them. Even in these noiseless simulations, no stable τ 0 was attained before the traces became too small to measure. It seems likely that this was also true of the full cell simulations. The spatiotemporal distribution of transmembrane voltage decay and the associated flows of equalizing currents in a fully branched neuron are complex in the presence of a somatic shunt and this also appears to be true in the cable surrogates. It seems important that the AT and DL equivalent cables mimic these factors more accurately than LM cables. Are These Equivalent Cables of Useful in Neuron Modeling? The primary purpose of the present work was to explore whether alternative morphoelectrotonic transforms can be used to reduce anatomically complex neurons into simple surrogate models that embody the electrotonic characteristics of the original cells more accurately than is possible in the LM cable formulation. The results presented answer this question affirmatively. Relatively coarse AT and DL cables mimic the steadystate and time-domain behaviors of complex neurons relatively well and may therefore be of use in network models that utilize dendritic properties in network elements. Such cables may also be useful in attempts to estimate specific membrane properties from morphological and electrophysiological data, which require time domain information (Rall et al., 1992). The possible presence of spatially nonuniform membrane properties greatly complicates such efforts (Clements and Redman, 1989; Fleshman et al., 1988; White et al., 1992). Prediction of transient behavior in realistically complex model neurons with nonuniform R m using analytical methods is neither straightforward (Major et al., 1993) nor entirely free of anatomical compromises (e.g., Evans and Kember, 1998; Poznanski, 1996). As a practical alternative, compartmental equivalent cables provide computationally efficient surrogates for use in trial-and-error membrane parameter estimation procedures (Clements and Redman, 1989). Such step-wise approaches also provide a heuristically

16 46 Burke satisfying way to explore sensitivity to changes in the various model parameters (Burke et al., 1994). In recent years, various alternative approaches have been suggested to simplify the electrotonic architecture of neurons into equivalent cables, some for specific purposes (e.g., Bush and Sejnowski, 1993; Douglas and Martin, 1992; Stratford et al., 1989) and others as heuristically interesting objects (Ogden et al., 1999; Whitehead and Rosenberg, 1993). A recent extensive theoretical analysis suggests that collapsing real neurons into equivalent cables according to electrotonic metrics that conserve membrane area, as in the present work, are superior to some of these alternatives (Ohme and Schierwagen, 1998). The AT and DL cable models discussed in this paper are specifically formulated to conserve the electrotonic distribution of membrane area. Construction of these cables, although seemingly complex (see Methods), is based on a straightforward premise that is easily understood. The program that generates them can reduce a digitized motoneuron with 700 to 1,000 compartments into an equivalent cable of any type in less than one second. The algorithm could easily be adapted to construct multicable cartoon representations of pyramidal neurons (Stratford et al., 1989). The new equivalent cable formulations not only reveal some interesting aspects of reduced neuron models, but they also appear to be practical alternatives to existing schemes. Acknowledgments The author wishes to thank Dr. Idan Segev for discussions that initiated this work and Dr. Hagai Agmon- Snir for his helpful comments on the application of the propagation delay metric to the construction of PD cables. References Agmon-Snir H (1995) A novel theoretical approach to the analysis of dendritic transients. Biophys. 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