Signal Transfer in Passive Dendrites with Nonuniform Membrane Conductance

Transcription

1 The Jornal of Neroscience, October 1, 1999, 19(19): Signal Transfer in Passive Dendrites with Nonniform Membrane Condctance Michael London, 1 Clade Menier, 2 and Idan Segev 1 1 Department of Nerobiology, Institte of Life Sciences and Center for Neral Comptation, Hebrew University, Jersalem 91904, Israel, and 2 Centre de Physiqe Théoriqe (UMR 7644 Centre National de la Recherche Scientifiqe), Ecole Polytechniqe, Palaisea Cedex, France. In recent years it became clear that dendrites possess a host of ion channels that may be distribted nonniformly over their membrane srface. In cortical pyramids, for example, it was demonstrated that the resting membrane condctance G m (x) is higher (the membrane is leakier ) at distal dendritic regions than at more proximal sites. How does this spatial nonniformity in G m (x) affect the inpt otpt fnction of the neron? The present stdy aims at providing basic insights into this qestion. To this end, we have analytically stdied the fndamental effects of membrane non-niformity in passive cable strctres. Keeping the total membrane condctance over a given modeled strctre fixed (i.e., a constant nmber of passive ion channels), the classical case of cables with niform membrane condctance is contrasted with varios nonniform cases with the following general conclsions. (1) For cylindrical cables with sealed ends, monotonic increase in G m (x) improves voltage transfer from the inpt location to the soma. The steeper the G m (x), the larger the improvement. (2) This effect is frther enhanced when the stimlation is distal and consists of a synaptic inpt rather than a crrent sorce. (3) Any nonniformity in G m (x) decreases the electrotonic length, L, of the cylinder. (4) The system time constant 0 is larger in the nonniform case than in the corresponding niform case. (5) When voltage transients relax with 0, the dendritic tree is not isopotential in the nonniform case, at variance with the niform case. The effect of membrane nonniformity on signal transfer in reconstrcted dendritic trees and on the I/f relation of the neron is also considered, and experimental methods for assessing membrane nonniformity in dendrites are discssed. Key words: cable theory; nonniform membrane condctances; dendritic ion channels; dendritic signal transfer; compartmental modeling; dendritic transients Transmembrane ion channels are the main carriers of electrical crrents in nerons. Their type, kinetics, and spatial distribtion may critically determine the properties of the electrical signals that are initiated and propagated in nerons. Importantly, these ion channels are distribted nonniformly over the neron srface. An obvios example is the myelinated axons where Ranvier nodes bear a high density of Na channels, whereas the internodes are bare of these channels (Hille, 1992). This spatial nonniformity has fnctional implications for the propagation speed of the action potential along the axon. Ion channels are distribted nonniformly also over the dendritic membrane. Are there some design principles that govern the distribtion of channels and optimize certain aspects of signal processing in dendrites? Early stdies on excitable properties of dendrites can be fond in Lorente de No (1947), Spencer and Kandel (1961), Llinas and Sgimori (1980), and Schwindt and Crill (1995); for review, see Received Febrary 8, 1999; revised Jne 16, 1999; accepted Jly 12, All analytical materials appear in the Appendixes on-line at This work was spported by the Keshet (Arc-en-ciel) France Israel exchange program and by the Israeli Academy of Science. We thank M. Rapp, J. C. Anderson, R. J. Doglas, K. A. C. Martin, and D. Trner for allowing s the se of their reconstrcted nerons. We thank D. Hansel becase the present work was initiated by a discssion with him on the Hoffman et al. (1997) paper. We are also indebted to K. Borejsza for carefl reading of this manscript. Correspondence shold be addressed to Michael London, Department of Nerobiology Institte of Life Sciences, Hebrew University, Jersalem, Israel Dr. Menier s present address: Laboratoire de Nerophysiqe et Physiologie d Système moter (EP 1848 Centre National de la Recherche Scientifiqe), Université René Descartes, Paris cedex 06, France. Copyright 1999 Society for Neroscience /99/ $05.00/0 Mel (1994). Recent experimental stdies sing infrared DIC video microscopy, combined with patch-clamp techniqes (Start et al., 1993; Dodt and Zieglgansberger, 1994) directly showed that, indeed, ion channels were distribted nonniformly over the dendritic srface (for review, see Johnston et al., 1996). Recordings from membrane patches, excised from the apical trnk of hippocampal CA1 pyramidal nerons, showed that both the A-type and the delayed rectifier K condctances linearly increased with the distance from the soma (Hoffman et al., 1997). The density of I h crrents also increase in these cells (Magee, 1998). An increase in the density of both I h crrent and the cesim-insensitive membrane condctance at distal dendritic regions was also recently observed in neocortical layer V pyramids (Start and Sprston, 1998). Nonniformity in dendritic excitability was also fond in other neron types (Start and Hässer, 1994; Trrigiano et al., 1995; Bischofberger and Jonas, 1997; Kavalali et al., 1997; for review, see Segev and Rall, 1998). Spatial distribtion and spatiotemporal activation pattern of transmitter-gated condctances may also case nonniformity of the dendritic membrane. Ths, the dendritic membrane condctance is expected to develop time- and activity-dependent spatial heterogeneities (Holmes and Woody, 1989). Most of or intitions regarding how the interplay between dendritic morphology, physiology, and inpt conditions determine the inpt otpt properties of dendrites rely on Rall s cable theory. In this framework, the effect of geometric nonniformities in the dendritic tree, inclding tapering and branching, was analyzed (Btz and Cowan, 1974; Horwitz, 1981, 1983; Poznanski,

2 8220 J. Nerosci., October 1, 1999, 19(19): London et al. Model of Passive Dendrites with Nonniform Membrane Table 1. Table of symbols V Transmembrane voltage (relative to resting potential) mv d Diameter of the dendritic cylinder cm Length of the dendritic cylinder cm x Physical distance from the origin (sally the soma) cm G m (x) Specific membrane condctance S/cm 2 G m 1/ 0 G m (x)dx Average membrane condctance S/cm 2 m (x) G m (x)/g m Dimensionless membrane condctance Dimensionless R m (x) Specific membrane resistance cm 2 R i Specific cytoplasm resistivity cm r i 4R i /( d 2 ) Internal resistance per nit length of cylinder /cm C m Specific membrane capacitance F/cm 2 Slope parameter ( 1 1) Dimensionless (x) d/4g m (x)r i Space constant (nonniform case) cm d/4g m R i Space constant (niform case) cm L 0 dx/ (x) Generalized electrotonic length Dimensionless L / Electrotonic length (niform case) Dimensionless X x/ Distance from location x to origin, in nits of the niform space constant Dimensionless R in (x) Inpt resistance at location x G in (x) 1/R in (x) Inpt condctance at location x S R x,y Transfer resistance from location x to location y I a Axial crrent A G ( /2) Gmd 3 /R i Inpt condctance of a semi-infinite niform cylinder S G L Axial leak condctance at bondary S B G L /G Bondary condition Dimensionless m (x) C m /G m (x) Local membrane time constant sec m C m /G m Membrane time constant (niform case) sec s n, n n-th eqalizing time constant (niform and slope case, respectively) sec (x), (t) Delta fnction (x, y, t) Green fnction g syn Synaptic condctance S E syn Synaptic reversal potential mv 1988; Schierwagen, 1989; Abbott, 1992; Holmes et al., 1992; Holmes and Rall, 1992a,b; Agmon-Snir and Segev, 1993; Major et al., 1993a c; Segev et al., 1995; Rall and Agmon-Snir, 1998), and the impact of nonlinear voltage- and transmitter-gated dendritic ion channels, as well as of massive synaptic inpt on the inpt otpt properties of nerons, was explored theoretically (Barrett and Crill, 1974; Shepherd et al., 1985; Miller et al., 1985; Fleshman et al., 1988; Segev and Rall, 1988; Clements and Redman, 1989; Holmes, 1989; Bernander et al., 1991; Rapp et al., 1992; Siegel et al., 1994; Bernander et al., 1994; De Schtter and Bower, 1994; Pinsky and Rinzel, 1994; Schwindt and Crill, 1995; Start and Sakmann, 1995; Wilson, 1995; Mainen and Sejnowski, 1996). A review of these isses can be fond in Koch (1999). Experimental stdies on the inpt otpt properties of nonlinear dendrites can be fond in Larent et al. (1993), Nicoll et al. (1993), Haag and Borst (1996), and Chen et al. (1997). However, there is yet no systematic stdy on the effect of membrane nonniformity in dendrites, althogh previos stdies did consider the possibility that the soma membrane is leakier than the dendritic membrane (Rall, 1962; Rall and Shepherd, 1968; Iansek and Redman, 1973; Drand, 1984; Kawato, 1984; Major et al., 1993a). Motivated by the recent experimental evidence abot membrane nonniformity in dendrites, the present stdy explores, sing analytical tools, whether there is some fnctional advantage of membrane nonniformity for the transfer of synaptic signals from the dendrites to the soma. In the tradition of W. Rall, we treat here, as a first stage, the case in which the dendritic membrane condctance G m (x) is spatially nonniform and passive. All mathematical considerations are groped in the Appendixes (online at MATERIALS AND METHODS The cable eqation (Rall, 1959) away from crrent sorces for a passive cylinder with nonniform membrane condctance is: 2 x 2 V x m x V V 0, (1) 2 t where m (x) C m /G m (x) is the (variable) passive membrane time constant and (x) d/(4g m (x) R i ) is the (variable) space constant, G m (x) is the specific membrane condctance, and d is the diameter of the cylinder. In the niform case, in which G m (x) is constant, (x), and m (x) m ; the sperscript, is sed throghot this work to label the niform case. These and other qantities are defined in Table 1. To nderstand the effects of different G m (x) fnctions, we need to impose some constraint that will allow s to compare these G m (x) fnctions. The most natral assmption is that the total membrane condctance of the cable is preserved in all cases: G m x dx const. (2) d 0 This implies that there is a fixed nmber of (passive) ion channels in the modeled strctre and that these channels are allocated to different regions of the cable, depending on the shape of G m (x).

3 London et al. Model of Passive Dendrites with Nonniform Membrane J. Nerosci., October 1, 1999, 19(19): The case of membrane nonniformity that will gain the main focs of the present work is the slope case, in which G m (x) linearly increases with the distance from the soma (x 0), namely, G m (x) ax b. This case has received some recent experimental spport (Hoffman et al., 1997). All qantities are labeled, in this case, with the sperscript s. In the case of cylinder of length, the constraint of a fixed total condctance, Eqation 2, implies that: G s m x G m x /2 1 2, (3) where G m is the average membrane condctance over the membrane area, x is the location along the cylinder (0 x ), and the dimensionless parameter a /(2G m ) is the slope a in nits of 2G m /. Becase G m (x) shold be positive for all x, is bonded, 1 1. The niform case is obtained when 0, whereas for the case of maximal slope, 1. In this case, the membrane condctance increases from 0 at x 0toamaximmof2G m at x. For DC crrent injection, the membrane voltage (away from crrent sorces) satisfies the steady-state cable eqation: 2 x d2 V V 0. (4) 2 dx Eqation 4 can be analytically solved for several nonniform condctance profiles. The soltions for typical increasing of G m (x) fnctions (power law, exponential, etc.) are given in Appendix B; they all involve special fnctions (Abramowitz and Stegn, 1970). Mathematica software (Wolfram, 1991) was sed for nmerical evalations of these analytical soltions; transient simlations and comptations of transfer resistances in reconstrcted dendrites were implemented sing NEURON simlator (Hines and Carnevale, 1997). The transient voltage response of a cable with nonniform membrane condctance can be compted sing the Green fnction, (x, y, t), of the problem which satisfies: ( ) 2 2 x, y, t x 2 m x, y, t G m x x, y, t x y t, t G m together with the imposed bondary and initial conditions. Separating time and space variables, the Green fnction can be expressed as an orthonormal expansion (Rall, 1969): x, y, t c j j x j y e j 0 t/ j, where the basis fnctions j satisfy: 1 0 j x k x dx 1 j k 0 j k. The system time constant 0 and the eqalizing time constants j (j 1), together with the associated eigenfnctions j ( j 0), are obtained by solving the eigenvale problem: ( ) 2 d2 j x dx 2 G m x G m m j j x 0, (5) with imposed bondary conditions at x 0 and x. It is meaningfl to distingish 0 from the other time constants (Rall, 1969). Indeed, 0 governs the slow relaxation of the voltage at large t, and for niform cables with sealed ends, 0 m. The other, smaller time constants control the faster spatial eqalization of voltage gradients, and they vanish in the limit of extremely compact cables (L 3 0). As shown in Appendix E, the eigenvale problem is analytically tractable in the slope case: the eigenvales are determined by solving a transcendental eqation, and the associated eigenfnctions can be expressed as linear combinations of Airy fnctions. The eigenvale Eqation 5 is formally identical to the Schrödinger eigenvale problem for the one-dimensional bonded motion of a particle in qantm mechanics (Landa and Lifschitz, 1997). This analogy, which is elaborated on in Appendix D.1, is sefl becase it provides a visal representation of the Green fnction of the cable problem in terms of discrete energy states in a potential well. It provides new insight on how the system time constant and the eqalizing time constants, together with the associated eigenfnctions, depend on the membrane condctance profile and on other cable parameters sch as the electrotonic length. Note also that another representation of the Green fnction, as a diffsion process, was proposed by Abbott et al. (1991), Abbott (1992), and Bressloff and Taylor (1993); this approach also relies on concepts and methods (i.e., path integrals or their discrete conterpart) that were previosly introdced in physics. These methods can be applied to any profile of nonniform specific condctance on the dendritic tree, bt they reqire the nmerical evalation of the contribtions of many paths. This limits its practical seflness mostly to the accrate estimation of the short-term transient behavior. In contrast, or approach is limited to certain types of condctance profiles, bt it provides analytical reslts and enables s to accrately determine, throgh the comptation of the larger time constants, how the slow relaxation of transients proceeds. RESULTS Steady crrent inpt in nonniform cylinders The steady-state voltage profile along a cylinder with both ends sealed, following constant crrent injection at different locations, is shown in Figre 1A for the niform case and the maximal slope case ( 1). This figre shows that the voltage at x 0 (soma) is always larger in the slope case, even when the local voltage response at the inpt site is smaller than in the niform case (e.g., at the distal end where x 1000 m). As demonstrated in Figre 1B, the voltage attenation between any inpt point x and the soma is smaller in the slope case. Note that this figre also describes the attenation of the time integral of the potential in the case of transient crrent injection (Rall and Rinzel, 1973). Ths, distribting the same total membrane condctance nonniformly rather than niformly improves the voltage response at the soma. To characterize the effect of steady crrent injection at point x on the voltage response at the soma, the transfer resistance R x,0 V(0)/I(x) is sed. Figre 2 A shows the normalized transfer resistance between the inpt location and the soma for cylinders of different electrotonic length. In the maximal slope case, the transfer resistance is always larger (for every x) than in the corresponding niform case, and ths for all inpt locations the voltage response at the soma is larger when the condctance linearly increases in the cylinder. This property is independent of the cylinder physical length. The difference between the two cases grows with the electrotonic length of the cylinder. It is also tre for all the intermediate linear slopes (i.e., 0 1). Figre 2B shows the normalized inpt resistance R in (x) R x, x for cylinders with electrotonic length L 1. In the niform case, R in (x) is symmetric with respect to the midpoint x /2, where the minimm is obtained. This is no longer tre in the maximal slope case in which R in (x) is larger in the proximal part of the cylinder and smaller in its distal part compared with the niform case. This is expected becase, in the slope case, the membrane is leakier in distal regions of the cylinder. Becase R in is determined not only by the local membrane condctance bt also by the cable properties and the bondary conditions, the intersection between R in (x) in the two cases is not at the cylinder s midpoint, X 0.5 (where G s m (x) G m ), bt rather at X A given crrent inpt at X 0.57 will reslt in a smaller local voltage response in the slope case becase of the smaller inpt resistance in these sites compared with that of the niform case. Nevertheless, the resltant voltage at the soma is larger in the slope case becase of the larger transfer resistance in this case. In contrast, for all X 0.57, both R in and R x,0 are larger in the slope case, giving rise to a larger voltage response both locally and at the soma. The benefit, in terms of the relative increase of the soma

4 8222 J. Nerosci., October 1, 1999, 19(19): London et al. Model of Passive Dendrites with Nonniform Membrane Figre 1. Voltage attenation is smaller in the slope case than in the niform case. A, Voltage profiles for a DC crrent injection at x 0, 100, 570, and 1000 m in the niform case (dotted line) and the maximal slope case (continos line; see inset). Voltage response at inpt locations (arrows) is marked by circles in the niform case and by crosses in the slope case. Cable parameters are 1000 m, d 4 m, R i 200 cm, R m 20,000 cm 2, and L 1 (the slope per nit length in the maximal slope case is 1E 5(S/cm 2 )/100 m). The vale of the DC crrent is chosen so that the local voltage response to crrent injection at x 0is 1 mv in the niform case. B, Log voltage attenation, LA x,0 ln(v(x)/ V(x 0)), from the inpt site, x, tox 0. voltage, gained from having a maximal slope condctance rather than a niform condctance ranges from 3% for distal inpts to 16% for proximal inpts in cylinders with electrotonic length, L 1 (see Fig. 4B, crrent inpt). The focs in the above analysis was on the slope case, in which G m (x) is a linear fnction of x, and more specifically on the limiting case where 1. In Figre 3, the transfer resistance for several power law fnctions of G m (x) with increasing exponent (0, 1/2, 1, and 2) is compted. All cases with integer power are actally analytically tractable (see Appendix B.2). G m (x) is normalized sch that the constraint of a fixed average membrane condctance (Eq. 2) holds, and in all nonniform cases, G m (0) 0. Figre 3 shows that, compared with the niform case, the transfer resistance is always increased by a gradient of membrane condctance, and the difference compared with the niform case increases with the steepness of the condctance profile. Indeed, when G m (x) increases with an exponent of 2, the benefit, in terms Figre 2. Voltage transfer is enhanced in cables with nonniform membrane condctance. A, Transfer resistance, R x,0, from inpt location x to the soma (x 0). The reslts are shown for three cylinders (L 0.5, 1, and 2, respectively) in the niform case (dotted lines) and the slope case (solid lines). B, Inpt resistance, R in, verss inpt location for the niform case (dotted line) and the slope case (solid line). L 1. Note that both R x,0 and R in are normalized by R, the inpt resistance of the niform cylinder when extended to be semi-infinite. As shown by Rall (1959), niform cylinders with the same L vale bt with different specific properties have the same transfer resistance, R X,0, when each cylinder is normalized by its corresponding R vale. This also holds tre for cylinders with linearly increasing membrane condctance, if the slope parameter and the electrotonic length L of the corresponding niform cylinder are the same for all cylinders considered. The normalizing factor is then the inpt resistance R of the semi-infinite niform cylinder of specific condctance G m G m. of the relative increase of the soma voltage, ranges from 6% for distal inpts to 26% at more proximal sites, with an average of 17%. Note also that becase of the reciprocity property of the transfer resistance [R x,y R y,x (Koch, 1999)], the effect of spatially decreasing condctance profiles can be dedced from these reslts (see Appendix C.3). Finally, we note that the generalized electrotonic length of a given cylinder, L 0 dx/ (x), is smaller for the slope case compared with the niform case. Indeed, for the slope case, L is given by: L L 1 3/2 1 3/2. 3 L decreases with increasing and reaches its minimm, L 8/9L, for 1. More generally, Appendix A.1 shows that

5 London et al. Model of Passive Dendrites with Nonniform Membrane J. Nerosci., October 1, 1999, 19(19): Figre 3. Transfer resistance increases with the steepness of G m (x). Transfer resistance is plotted for a cylinder of length L 1 for different condctance profiles; niform, G m (x) G m (dotted line); sqare root, G m (x) G m (3/2 x/ ) (dotted dashed line); linear, G m (x) G m (2x/ ) (continos line); sqare, G m (x) G m (3(x/ ) 2 )(dashed line). The corresponding G m (x) fnctions are drawn in the inset. any membrane condctance heterogeneity decreases L in comparison with the corresponding niform case, making the cable electrotonically more compact. However, this does not necessarily imply that voltage attenation is redced in these more compact cables (see Discssion). Synapses as inpt Realistic synaptic inpts involve condctance changes, which implies that the synaptic crrent nonlinearly depends on the synaptic condctance change g syn. The difference in inpt resistance between the slope and the niform case (Fig. 2B) leads to a different degree of local voltage satration in these two cases, and conseqently, to differences in the amont of crrent generated by the synapse. The voltage response at x 0, attribtable to the steady-state activation of a synapse at location x is: V 0 g syn E syn G in x g syn G in x R x,0, (6) where E syn is the reversal potential of the synaptic crrent, and G in (x) 1/R in (x) is the inpt condctance at location x before the activation of the synapse, as compted in Figre 2B. Note that the right side of the eqation is simply composed of the local inpt crrent prodced by the synapse, mltiplied by the transfer resistance from the inpt site to the soma. The derivation of this eqation is given in Appendix C.2. Figre 4A shows the voltage at x 0 in response to the steady-state activation of an excitatory synapse in the maximal slope case and the niform case (bottom two crves). As a reference, the voltage response to a DC crrent inpt, I g syn E syn, is also shown (top two crves). This is the crrent that the synapse wold generate if the synaptic crrent were not limited by satration, and this is, therefore, the pper bond on the synaptic crrent that is actally generated. The corresponding crves are then proportional to R x,0 (compare with Fig. 2A). As expected from synaptic satration, V(0) in both the niform case and slope case is smaller for the condctance inpt case compared with the corresponding crrent inpt case. Still, as for the crrent inpt, the soma voltage is larger in the slope case for all inpt locations Figre 4. Distal synaptic inpts are more efficient when G m (x) linearly increases with distance. A, Soma voltage as a fnction of inpt location in the cylinder for both a steady-state synaptic condctance change (g syn 2 ns, E syn 65 mv; bottom two crves) and for a DC crrent injection (I g syn E syn ; top two crves). The niform case (dotted line) and the maximal slope case (continos line) are shown; cable properties are as in s Figre 1 A. B, The relative increase in soma voltage, (V soma V soma )/ V soma, denoted as Benefit, attribtable to the spatial gradient in the membrane condctance, is displayed as a fnction of inpt location. Crrent injection and three cases of steady-state synaptic condctance change (g syn 1, 10, and 100 ns) are shown as a fnction of inpt location. The dashed crve corresponds to the limiting case in which g syn 3. This crve was compted by taking the limit, g syn 3, in Eqation 6. The relative increase in soma voltage is then given by (AF x,0 s s AF x,0 )/AF x,0. This can be interpreted as the benefit attribtable to the difference in the attenation factor. This is reasonable becase, in this limit, the local voltage is E syn in both the slope and the niform cases, and only the difference in AF x,0 cases a difference in soma voltage. (Note that in this limit, the soma voltage is inversely proportional to the attenation factor.) (Fig. 4A, two bottom crves). To highlight the effect of the nonniform membrane condctance in the case of synaptic inpts, we have plotted in Figre 4B the benefit (the relative change in the soma voltage) of having a maximal slope membrane condctance rather than a niform membrane condctance. The reference case of a DC crrent inpt, I g syn E syn, is also shown (crrent inpt crve); the corresponding crve is independent of s g syn (becase of linearity) and is given by (R x,0 R x,0 )/R x,0. All of the crves in Figre 4B intersect at X At this site, the inpt resistance in the slope case is the same as in the

6 8224 J. Nerosci., October 1, 1999, 19(19): London et al. Model of Passive Dendrites with Nonniform Membrane Table 2. Bondary condition (B) at the soma for the dendritic trees in Figre 7 Cell type Dendrite B 20,000 B 50,000 d ( m) max L 20,000 Layer V pyramidal Apical Basal (n 10) Hippocampal CA1 Apical Basal (n 6) Cerebellar Prkinje B vales were compted at the point of connection between the soma and the dendrite indicated (e.g., the apical dendrite). Comptations were performed assming niform R m of 20,000 cm 2 (third colmn) or 50,000 cm 2 (forth colmn). In both cases R i 200 cm. For each dendrite B (2/ ) R m R i d 0 3/2 /R* in, where d 0 is the diameter of the dendrite at the connection point with the soma (fifth colmn), and R* in is the inpt resistance of the neron at the soma when the dendrite is removed. The last colmn denotes the longest electrotonic path, from the soma to a dendritic terminal, for the niform case (R m 20,000 cm 2 ). niform case (Fig. 2B), and the difference between the two cases (which is also independent of g syn ) is attribtable only to differences in the transfer resistance, R x,0 (Eq. 6). Most notable is that the relative change in the soma voltage is always positive (and is independent of E syn ). This means that when the inpt is a synaptic condctance change, the nonniformity in the membrane condctance reslts in an increase of the soma voltage compared with the niform case, as was also the case for crrent inpt. For inpt locations more distal than the point of intersection, the crrent inpt case sets a lower bond for this effect, and the benefit of having a slope membrane condctance grows with increasing g syn, eventally reaching a limit (Fig. 4B, dashed crve). In contrast, for proximal inpt locations, the crrent inpt case sets an pper bond for the effect of the slope membrane condctance. The benefit of having a slope condctance is smaller when the inpt is a synapse rather than a crrent inpt, and this benefit decreases with increasing g syn. These reslts can be explained as follows. First, in the slope case, more synaptic crrent is generated at leakier distal inpts sites becase the local voltage satration is redced as a reslt of the lower inpt resistance at these sites (Fig. 2B). The opposite is tre for proximal inpt sites. Second, the transfer resistance is always larger in the slope case compared with the corresponding niform case (Fig. 2A). All in all, distal synaptic inpts benefit twice from having slope membrane condctance. For proximal synaptic inpts, however, there is a competition between the redction of the synaptic crrent on the one hand (cased by stronger satration) and the increase in the transfer resistance on the other hand. Becase the latter dominates, synapses are always more effective in prodcing a larger soma voltage when G m linearly increases with distance. Distal synapses can exploit this membrane nonniformity more effectively, becase of a decreased synaptic satration, than do proximal synapses. As a conseqence, the largest relative benefit is obtained for weak proximal synapses and for strong distal synapses. Dendritic trees The previos sections dealt with cylinders of constant diameter with both ends sealed. There are two main difficlties in extending the insights gained in these sections to dendritic trees with realistic geometries. The first arises from the sealed ends assmption and the second from the assmption of constant diameter. Real dendrites are complex branching strctres with freqent changes in diameters, and althogh sealed end bondary condition is sally accepted for the termination of distal dendritic arbors, this condition is generally inappropriate at the proximal end of dendrites, toward which the synaptic crrent flows. At this end, the impedance load attribtable to the soma and the other dendritic trees emerging from it may reslt in leaky bondary conditions. In the following, we will separately deal with each of these isses and then explore their interplay by considering relevant models of reconstrcted nerons. Bondary conditions When the synaptic crrent flows from the inpt dendrite toward the soma, the steady-state bondary condition at the soma is given by the leak condctance, G L, which is the sm of the individal inpt condctances of all the other dendrites, each taken alone, pls the inpt condctance of the somatic membrane (Rall, 1959). The leakiness of the bondary condition can then be qantified by the parameter B G L /G, where G is the inpt condctance of a semi-infinite cylinder with the same diameter and specific properties as the inpt dendrite. B 0 is the sealed end bondary condition (no leak throgh the termination), and B is the killed end condition. Large B vales indicate a large leak at the bondary. In Table 2, B was compted for three reconstrcted dendritic trees shown in Figre 7, assming niform membrane resistance of 20,000 cm 2 (forth colmn) and 50,000 cm 2 (fifth colmn). The vales obtained span two orders of magnitde, from 0.1 (in Prkinje cells) to more than 10 for the (thin) basal dendrites of hippocampal and neocortical pyramidal cells. For these basal dendrites, the soma and all other dendrites impose a large condctance load. The bondary conditions for the apical tree of these two cells are less leaky. How do leaky bondary conditions affect the reslts obtained ths far? Figre 5A depicts the effect of varying bondary conditions at x 0 on the transfer resistance R x,0. When the proximal termination becomes leaky (B 1), the transfer resistance crves in the niform case and in the slope case intersect; the intersection point moves closer to the proximal end as B increases. In contrast to the sealed end case, in these cases distal crrent inpt gives rise to a smaller somatic voltage in the slope case compared with the niform case. The attenation factor, however, remains smaller in the slope case (Fig. 5B), even with large B vale, althogh the difference between the two cases progressively diminishes as the leakiness at the termination increases. The reslts of Figre 5A, B demonstrate that, with increasing B vales, the longitdinal crrent flow in the cylinder becomes more and more dominated by the leakiness at the bondary. Conseqently, the effects of the membrane nonniformity become progressively less significant. This effect is frther demonstrated in Figre 5C. Here a DC crrent is injected to the distal end of the cylinder, and the axial crrent I a (given by

7 London et al. Model of Passive Dendrites with Nonniform Membrane J. Nerosci., October 1, 1999, 19(19): in the corresponding niform case. This is tre for this specific distal inpt location and is the reslt of the larger crrent loss (shnt) throgh the leakier distal membrane in the slope case. One may erroneosly conclde from this figre that the charge transfer from x to the soma (x 0) is always smaller in the slope case for all inpt locations and bondary conditions. However, for many inpt locations, V soma is larger in the slope case (Fig. 5A), which implies that the charge transfer to the soma is then larger than in the niform case. This is tre becase I a (soma) V soma G L, and G L is the same in the niform case and the slope case (same B vale). Hence, the charge that reaches the soma can be easily derived by scaling Figre 5A by the appropriate G L vale that corresponds to a given vale of B. Then, for a given B vale and inpt location x, the axial crrent that reaches the soma is s larger in the slope case if R x,0 R x,0. In the figre above we imposed the same leak condctance (G L ) at the bondary for both the slope case and the niform case. However, if we impose the slope condition on a realistic dendritic tree, we expect that for each dendrite, B will be different from its vale in the corresponding niform case. This is becase the inpt resistance at the somatic end is increased in the slope case compared with the corresponding niform case (Fig. 2), and ths G in into this dendrite is decreased. This decreases G L for all the other dendrites (and ths the B vale becomes smaller). Note that Figre 5A also shows that in both the niform and slope cases, the smaller B is, the better the voltage transfer from the dendrites to the soma. This means that a dendrite with a slope condctance gradient will impose a lesser condctance load to the other dendrites stemming from the soma, compared with a dendrite with niform membrane condctance. Signal transfer from these other dendrites to the soma will then be improved becase the bondary condition at their somatic end will be less leaky. To assess what are the effects of this interplay between condctance gradients and bondary conditions, it is necessary to consider reconstrcted nerons (see below). Figre 5. Leaky bondary conditions redce the effects of membrane nonniformity. A, Transfer resistance R x,0 (x); B, log voltage attenation (LA x,0 ); C, axial crrent along the cylinder in response to a DC crrent injection at x. Reslts are displayed for a cylinder of length L 1, with sealed end at x and for three different bondary conditions at x 0(B 0.1, B 1, and B 10). Both the niform case (dotted lines) and the slope case (solid lines) are shown. 1/r i V/ x) is plotted as a fnction of x for three bondary conditions. First, notice that in both cases (slope and niform), the larger the B vale, the larger is the axial crrent I a. Note also that the axial crrent in the slope case is everywhere smaller than Branching The reslts derived above for cylinders cannot be readily extended to branching strctres. Consider the simple case of a branched cable consisting of a father branch and two daghter branches, one thick and one thin, sch that this strctre is eqivalent to a single cylinder in the niform case (Rall, 1959). This implies that the thick daghter branch is physically longer than the thin daghter branch. Now consider the maximal slope case in which the specific membrane condctance increases with a constant slope per nit length from the proximal end of the father branch to the distal terminals of the daghter branches. This case is analytically handled in Appendix A.2 and illstrated in Figre 6. Becase the thin branch is physically shorter (and the condctance is growing as a fnction of distance from the soma), the total membrane condctance in this thin and shorter branch is smaller than in the thicker sibling branch, and the membrane at its termination is less leaky. Indeed, most of the membrane condctance is now allocated to the thick branch. One conseqence of this asymmetry in the allocation of ion channels is that the branched strctre is no longer eqivalent to a single cylinder in the slope case; the thin branch is now electrically shorter than the thick branch. Another conseqence is that the total condctance along a given path from the soma to some terminal is not the same in the niform case and the nonniform case. Ths, when comparing the electrotonic properties of this strctre between the niform and slope case, the

8 8226 J. Nerosci., October 1, 1999, 19(19): London et al. Model of Passive Dendrites with Nonniform Membrane second (very thin and short) daghter branch now has negligible membrane condctance. A branched dendritic strctre is generally not eqivalent to a cylinder even in the niform case, and in principle, the transfer resistance from distal sites to the soma may then be smaller in the slope case than in the niform case. An example is shown in Figre 6B in which the thin branch is physically longer than the sibling thick branch (see inset), so that the specific condctance of its terminal is larger than in the other branch. The transfer resistance for distal inpt locations on this branch steeply decreases with x, and it reaches lower vales in the distal part (arrow) in the slope case compared with the niform case. This effect is observed in the models of reconstrcted dendritic trees analyzed below, where some of the distal branches are mch longer than others. It is important to note that other rles for increasing membrane condctance cold be implemented on dendrites displaying geometrical nonniformities sch as branching, tapering, or flare. For instance, a reasonable alternative is that the specific membrane condctance increases with distance as a fnction of the membrane area rather than with distance (i.e., G m (x) 0 x d(z)dz). Another possibility is that the membrane condctance increases as a fnction of the electrotonic distance from the soma (i.e., G m (x) 0 x dz/ (z)). In the latter case, a branching strctre that is eqivalent to a single cylinder in the niform case is also eqivalent to a cylinder in the nonniform case. Figre 6. Signal transfer in nonniform branching geometries. Transfer resistance R x,0 is plotted as a fnction of inpt location, x, in a branched strctre. The niform case (dotted lines) and the maximal slope case (continos line), in which G m (x) linearly increases with the same slope along all the branches, are shown. The geometry of the branched strctre (inset) is as follows. A, 500 m and d 4 m for the father branch; for the thin daghter branch, 316 m, d 1.6 m, and for the thick daghter branch, 453 m, d 3.3 m. In the niform case this strctre is eqivalent to a single cylinder of length L 1. B, Same as in A, bt the thin daghter branch is 200 m longer (see inset). In both A and B the constraint of a fixed total membrane condctance (Eq. 2) was imposed. This yields a slope condctance of 1.042E 5(S/cm 2 )/100 m in case A and 0.967E 5(S/cm 2 )/100 m in case B. insights gained from a single cylinder are not directly valid becase the constraint of the total amont of condctance imposed on the whole strctre is not satisfied for each path separately. Becase of the asymmetry in the electrotonic properties of the two daghter branches in the slope case, the transfer resistances from the terminals of these branches to the soma are no longer eqal as they were in the niform case. Figre 6A shows that R x,0 from distal sites in the thin branch is larger than R x,0 from the corresponding sites of the thick branch. Still, R x,0 is larger in the slope case than in the niform case, whatever the inpt location x is, as was the case for single cylinders. This is tre even when the asymmetry between the two daghter branches is very strong. In that case, the diameters of the father branch and of the thick daghter branch are nearly eqal, and these two branches almost constitte a single cylinder of constant diameter, whereas the Reconstrcted trees The net effect of membrane nonniformity on the transfer resistance for three reconstrcted dendritic trees (all known to have active, and spatially nonniform, dendritic properties) is shown in Figre 7. In all cases, G m 20,000 S/cm 2. In the slope cases, the membrane condctance linearly increases with the physical distance from the soma, bt the total amont of condctance was kept eqal to the niform case (sch that the average vale of membrane condctance, G m, compted with respect to membrane area, was eqal to G m ). We first consider the case of the cerebellar Prkinje cell (Fig. 7A) where only one profse dendritic tree stems from the soma. As can be seen, the transfer resistance from most of the dendritic locations to the soma is larger in the slope case (heavy line). For this neron, the bondary condition at the soma end is close to a sealed end (Table 2); this is a favorable condition for the enhancement of signal transfer by the condctance gradient (see above). However, notice that becase this cell is electrically very compact, the net effect of membrane nonniformity is rather small (Fig. 2A, L 0.5; Table 2, last colmn). We next consider a neocortical pyramidal cell (Fig. 7B) where a clear distinction exists between short basal dendrites and an apical tree, with a large and thick trnk and a distal tft. In the slope case (heavy line), G m (x) linearly increases with the same slope in all dendrites. In the basal trees, the transfer resistance is almost twice as large in the slope case compared with the niform case. R x,0 is also larger in the slope case for the main trnk of the apical tree. In the apical tft, however, R x,0 is lower compared with the niform case. This is becase of the combination of the higher total G m of the apical dendrite in the slope case (prodcing a larger shnt, on the average) and the condctance gradient along this dendrite that prodces a strong local shnt at distal sites. As a conseqence, distal inpts are less efficient in this case compared with the niform case. Note that this example encapslates many of the points discssed so far. The bondary condi-

9 London et al. Model of Passive Dendrites with Nonniform Membrane J. Nerosci., October 1, 1999, 19(19): Figre 7. Voltage transfer is enhanced in dendritic trees with slope membrane condctance. Transfer resistance in the slope case and the niform case for three reconstrcted trees: A, cerebellar Prkinje cell (provided by M. Rapp, Hebrew University); B, D, layer V neocortical pyramidal cell (provided by J. C. Anderson, K. A. C. Martin, and R. J. Doglas, ETH, Zrich); C, hippocampal CA1 cell (provided by D. Trner, Dke University). D, Same as B bt now the total condctances of the apical tree and of the basal dendrites both retain separately in the slope case the vales they have in the niform case. In A C, the constraint of a fixed total membrane condctance in slope and niform cases was imposed for the whole dendritic arborization. In all for cases G m 20,000 S/cm 2 and R i 200 cm. To keep the constraint of a fixed total membrane condctance, a slope of 2.5E 5(S/cm 2 )/ 100 m was sed in A, 1.5E 5(S/cm 2 )/ 100 m in B, and 8.6E 6(S/cm 2 )/100 minc.indaslope of 9.2E 6(S/cm 2 )/ 100 m was sed for the apical dendrite, and a slope of 4.5E 5(S/cm 2 )/100 m was sed for all the basal dendrites. Scale bar, 100 m. tions at the soma end are leaky (Table 2), there is a reallocation of condctance from the shorter basal dendrites to the longer apical tree, and there are long paths in this tree where the specific membrane condctance becomes very high. The same general behavior also holds tre for the hippocampal CA1 neron (Fig. 7C), bt the advantage of having slope condctance is also apparent at distal apical arbors. We shold emphasize that the reslts shown in Figre 7 are for steady crrent inpt; with a synaptic condctance change, this effect is expected to be even larger, especially in thin arbors in which local synaptic satration may become significant (Fig. 4B). The enhancement of signal transfer in basal dendrites of pyramidal cells on Figre 7B, C is cased mainly by the reallocation of the condctance in the slope case from the short basal tree to the long apical tree. This effect is analyzed on a very simple model in Appendix A.2. To control for the effect of condctance reallocation from one sb-tree to the other (e.g., basal to apical), we model in Figre 7D the case in which each of these sb-trees conserves its own total condctance as it has in the niform case. Note that this implies that the slope per nit length is different between the two sb-trees (see inset). The enhancement of voltage transfer in the basal tree is now drastically redced compared with Figre 7B. For the same reason, the difference between the slope and niform cases in the hippocampal neron (Fig. 7C) is less dramatic than in the layer 5 pyramid. Indeed, the asymmetry between the apical and basal tree of the CA1 neron is smaller, and ths, the reallocation of G m is less marked in this case. As shown in Table 2, the B vales at the soma are qite different between the basal and apical tree. For the basal trees, a large condctance load is imposed by the soma and apical dendrite (B 10), especially when condctance reallocation to the apical tree is allowed (Figs. 7B, C). Becase the effect of gradients in membrane condctance diminishes with large B vales, one expects that nonniformity in membrane condctance will have a relatively small effect for the basal tree. This was verified by comparing the case where the condctance linearly increases in the basal tree with the case in which the same total condctance in the basal tree is niformly distribted (reslts not shown). For the apical tree, however, the soma and the basal tree impose a mch smaller condctance load (B vales are smaller), especially if condctance reallocation between the basal and apical trees is allowed. This explains why gradient membrane condctance can improve signal transfer in the apical tree. Transient crrent injections In the above we dealt with the steady-state case, which is also applicable to the behavior of the time-integral of transient voltages (Barrett and Crill, 1974; Rinzel and Rall, 1974). In this section we consider transient crrent injection I(x, t) and restrict orselves to the case of a single cylinder. The soltion of the time-dependent cable eqation, which linearly depends on the crrent inpt I(x, t), can be obtained by convolving this crrent inpt and the Green fnction (x, t)ofthe problem. (x, t) is the voltage response at x, t to a crrent plse at time t 0 and location x x in. In the niform case, (x, t) can be written as a infinite sm of fnctions, which exponentially decay with time: x, t 0 c j j e t/ j. The coefficients, c j, are given by the expansion of the crrent implse I(x, t) (x x in ) (t) on the set of orthonormal fnctions j, so that c j j (x in )/ (Rall, 1969). For sealed ends

10 8228 J. Nerosci., October 1, 1999, 19(19): London et al. Model of Passive Dendrites with Nonniform Membrane Figre 8. Voltage transients in a cylinder with nonniform G m. The voltage profile V(x) along a cylinder (L 1.5, m 20 msec) is plotted as a fnction of the dimensionless variable X x/ at sccessive times (t 1, 5, 11, 20, 80 msec), after a brief crrent injection at X Both the niform case (dotted line) and the maximal slope case (solid line) are shown. The right two graphs show the voltage response at X 1.05 (large initial transient) and at X 0.45 as a fnction of time for both the niform case (bottom) and slope case (top). The arrows mark the times at which V(x) is displayed on the left. bondary conditions, the system time constant 0 is eqal to the membrane time constant, m C m /G m, and the associated fnction, 0 1, is spatially constant. The smaller eqalizing time constants are given by: j m 1 j 2 L The associated fnctions j cos( j x/ ) 2 are delocalized over the whole interval [0, ]; only their phase varies with the position x, whereas their amplitde remains constant. This stems from the niformity of the cable that precldes that any particlar spatial location be privileged. Figre 8 (left) displays the voltage profile along a cable of intermediate electrotonic length (L 1.5) at sccessive times, after a brief crrent injection. It highlights the striking differences between the maximal slope case and the niform case. At time t 1ms, after the onset of the inpt crrent, the voltage profiles are essentially similar in the niform case and maximal slope case (Fig. 8, t 1 ms), bt once the asymptotic regime is reached, where the decay of transients is governed by 0, the voltage profile always remains nonniform in the slope case, whereas the cable is isopotential (p to exponentially small corrections) in the niform case (Fig. 8, t 20 and t 80). The behavior of the transients at right is elaborated in Discssion. This nonniformity reflects the behavior of 0 (x) in the slope case, which decreases from the proximal low condctance region to the distal high condctance region, as is proved in Appendix E.3. Moreover, the final relaxation phase proceeds more slowly in the nonniform case. These differences can be nderstood by stdying the Green fnction s (x) of the slope case. An expansion of time-dependent soltions as a sm of orthogonal fnctions that decrease exponentially in time can also be derived in the nonniform case. The time constants, j, and the associated eigenfnctions, j, are soltions of the eigenvale problem (Eq. 5), which can be recast into the dimensionless form: L 2 d2 j y dy 2 m y m j j y 0, where y x/ and m (y) G m (y)/g m. The normalization of j then becomes 0 j 2 (y)dy. In the slope case, this eigenvale problem is analytically tractable (see Appendix E). The eigenvales and eigenfnctions depend not only on the condctance profile, m (y), bt also on the electrotonic length, L, which plays the role of a diffsion constant in the above eqation and controls the degree of spatial averaging. In particlar, fll spatial averaging occrs in the limit L 3 0, where the diffsion length becomes large with respect to the length of the cable, so that the time constants tend to the vales obtained for the niform case, whatever the actal condctance profile along the cable is. For the benefit of the physics-oriented reader, the effects of L are discssed in Appendix D.1 sing a formal analogy with qantm mechanics. In Figre 9A, the system time constant, 0 s, and the first two eqalizing time constants, 1 s and 2 s, are plotted in the maximal slope case and for sealed ends bondary conditions, as a fnction of L in the range 0 L 10. This range shold cover all physiologically relevant sitations, even when strong backgrond synaptic activity increases the effective electrotonic length of the dendritic tree (Bernander et al., 1991; Rapp et al., 1992). In this range, the system time constant 0 in the slope case is larger than in the corresponding niform case. This is actally tre for all vales of L, and not only in the slope case, bt also whenever the membrane condctance is not niform (as proved in Appendix D.2). For the general slope case, it can be shown that conversely 0 m /(1 ) (see Appendix E.3). Moreover, 0 s monotonically