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Democratization in a passive dendritic tree: an analytical investigation

One way to achieve amplification of distal synaptic inputs on a dendritic tree is to scale the amplitude and/or duration of the synaptic conductance with its distance from the soma. This is an example of what is often referred to as “dendritic democracy”. Although well studied experimentally, to date this phenomenon has not been thoroughly explored from a mathematical perspective. In this paper we adopt a passive model of a dendritic tree with distributed excitatory synaptic conductances and analyze a number of key measures of democracy. In particular, via moment methods we derive laws for the transport, from synapse to soma, of strength, characteristic time, and dispersion. These laws lead immediately to synaptic scalings that overcome attenuation with distance. We follow this with a Neumann approximation of Green’s representation that readily produces the synaptic scaling that democratizes the peak somatic voltage response. Results are obtained for both idealized geometries and for the more realistic geometry of a rat CA1 pyramidal cell. For each measure of democratization we produce and contrast the synaptic scaling associated with treating the synapse as either a conductance change or a current injection. We find that our respective scalings agree up to a critical distance from the soma and we reveal how this critical distance decreases with decreasing branch radius.


Introduction
There is experimental evidence that synaptic efficacy increases as a function of distance from the soma (Iansek and Redman, 1973;Andersen et al., 1980;Jack et al., 1981;Magee and Cook, 2000;Andrasfalvy and Magee, 2001;Migliore et al., 2005;Nicholson et al., 2006b), at least in CA1 hippocampal cells and spinal motoneurons. This is most likely due to increases in receptor density and/or vesicle release probability. One way to achieve such an increase in synaptic efficacy is through local anti spike timing-dependent plasticity (Rumsey and Abbott, 2004;Rumsey and Abbott, 2006). Distance dependent synaptic scaling is just one of several mechanisms for achieving so-called "dendritic democracy" whereby the spatially extended single neuron can compensate for dendritic attenuation (London and Segev, 2001). Other mechanisms for boosting somatic response to distal inputs include subthreshold resonance (via active currents such as I h and I A ), local dendritic spikelets, and global dendritic spikes. For a comprehensive review of "dendritic democracy", rendering excitatory postsynaptic potential (EPSP) amplitudes at the soma insensitive to dendritic origin, we refer the reader to Häusser (2001).
Our work is concerned with the analytical derivation of synaptic scaling rules that achieve dendritic democracy in purely passive models of branched dendrites. In particular, we ask: How must a synapse vary with distance to the soma in order that its impact on the somatic voltage be independent of its location? To make this question precise we need to model shunting synaptic currents. Postsynaptic shunting currents are induced by localized conductance changes associated with specific ionic membrane channels. The resulting currents are generally not proportional to the input conductance changes, and the conversion from conductance changes to membrane potential response is a nonlinear process. We model this as the product of a conductance, with amplitude g σ and duration T , and a shunt, measuring the difference between the membrane voltage v and the synaptic reversal potential E (Fatt and Katz, 1951;Burke, 1957;Jack et al., 1975;Tuckwell, 1988b).
There is common agreement that the efficacy of a synapse is determined by the amplitude and duration of the resulting somatic depolarization (see (Agmon-Snir, 1995) for a detailed discussion). We quantify the impact of synaptic input via the 5 measures of somatic response illustrated in Fig. 1: 1) the strength, V 0 , defined as the area under the graph of the somatic depolarization, 2) the centroid, C, of this area, 3) the width, W , of this area, 4) the peak depolarization, v max , and 5) the time, t max , at which this peak occurs. The first three of these measures are moment based and discussed more fully in (Agmon-Snir, 1995;Rall and Agmon-Snir, 1998), while the peak amplitude is used, for instance, in (Magee, 2000). The peak, v max , captures the amplitude, C, W and t max speak to duration, and the strength, V 0 , is a composite of amplitude and duration.
Presuming the soma to lie at x = 0, and the synapse to be located at x σ , the democratization problem now reduces to scaling g σ and T with somatic (x = 0) potentials. The latter is shorter and broader than the former on account of the cable filter. We quantify the somatic potential in terms of the strength, or area, V0, centroid, C, width, W , peak, vmax, and time to peak, tmax, of the cross-hatched region.
x σ in such a way that (combinations of) the aforementioned measures of the soma potential do not vary with x σ . We find that for x σ close to the soma this scaling, with respect to g σ , is linear, whilst further away it increases faster than linear. Interestingly, beyond a critical distance there is no choice of conductance strength that can lead to democracy.
Typically the mathematical treatment of shunting currents in dendritic systems is more involved than that of current injection, and is often abandoned in favor of a numerical analysis. Indeed previous work on synaptic scaling has relied upon compartmental modeling -see, for example, (London and Segev, 2001;Golding et al., 2005;Nicholson et al., 2006a). Here we discuss two mathematical approaches that circumvent the need for numerical simulations. The first technique equalizes the impact of spatially distributed synapses as measured by the moments of the somatic response (the first three measures of synaptic impact above). While this method is exact, its use in equalizing peak amplitude, that is measure 4), leads to a mathematically intractable problem. We therefore turn to a truncated Neumann series to construct the somatic response to shunting synaptic input. Even at second order we find excellent agreement with numerically obtained solutions of the full model within a physiologically realistic parameter set. We also contrast our findings with results achieved under the simpler hypoth-esis that synaptic activation results in current injections rather than conductance changes.
We start by a discussion of the cable models in §2. We describe the approach using moment based measures in §3 by constructing exact solutions to moments of the appropriate cable equation. Next, in §4, we show how to solve the model with shunts in terms of an infinite Neumann series. The first term of this expansion recovers the model without shunts (i.e. a simple current injection). A comparison of the series truncated at first and second order with numerically obtained solutions allows us to delimit their regimes of validity. Dendritic democracy which equalizes peak somatic response is discussed in §5. Finally, in §6, we discuss natural extensions of the work in this paper.

Cable models and preliminary analysis
The transmembrane potential, v(x, t), relative to rest, at location x and time t along an infinite uniform passive cable with a synapse at x = x σ , satisfies (Tuckwell, 1988b;Koch, 1999) where the cell is initially at rest, i.e., v(x, 0) = 0 for all x, and a is the radius of the cable, R i is the specific cytoplasmic resistivity, C m is the membrane capacitance, and G L is the membrane conductance.
Here H is the Heaviside function, and δ denotes the Dirac-delta function, so that the synaptic current or shunting current is the product of a synaptic conductance, of amplitude g σ and duration T , and a driving force biased by the reversal potential (relative to rest), E. If one ignores the influence of the local potential, v, on the synaptic current, then one arrives at (2) the model for direct current injection.
Clearly, the analysis of the autonomous Eq.
(2) is significantly easier than that of the non-autonomous Eq. (1). Throughout this work we identify the parameter regime in which model (2) is a good approximation of (1), as well as when this approximation breaks down.
In order to solve Eq. (1), we note that, away from the synapse, v satisfies while at the synapse, x = x σ , we have In the absence of end effects we expect the two one-sided slopes, v x (x ± σ , t), to be equal and opposite, and, as we assume that 0 < x < x σ , we write Eq. (4) as For infinite T we follow (Tuckwell, 1988b) and evaluate the Laplace transform of Eqs.
(3)- (5), where γ is the nondimensional synaptic conductance We next invert (6) and arrive at the voltage at the synapse (8) In the case that T is finite we note that, as Eq. (1) has but one solution, the expression in (8) is the solution of Eq. (1) up to time T . Although we shall see in §4 that the solution for times greater than T can be expressed in terms of a convolution of (8) and the free-space Green's function, it turns out that the exact moments of t → v(x σ , t) require only knowledge of (8) up to time T . Finally we recall that the Green's function leads to an explicit expression for u, the solution of the direct current injection model, Eq. (2). In particular, with t ∧ T ≡ min{t, T }, (Tuckwell, 1988a) where A(y, t) = 1 2 exp(−y/λ)erf y/(2λ) τ /t − t/τ − exp(y/λ)erf y/(2λ) τ /t + t/τ .

Moment Methods
We recall that synaptic impact or efficacy upon the soma is typically captured by some measure of the amplitude and duration of the resulting somatic potential. We start by following (Agmon-Snir, 1995) in choosing moment based measures. To begin, we denote the n-th moment in time of the potential v(x, ·) at x by Recalling Fig. 1 we focus on three common combinations of the first three moments: − the strength is simply the zeroth moment, , is the square of the width, W .
Beginning with the first, we construct an exact expression for the somatic strength, V 0 (0), in terms of the synaptic strength V 0 (x σ ). We then derive an exact expression for the latter in terms of the nondimensional synaptic amplitude, γ (see (7)) and nondimensional synaptic duration We then show, for fixed η, how to scale γ with x σ , up to a critical distance x lim σ , so that the somatic strength remains at a specified value V 0 . We conduct similar analysis of both C(x) and D(x) and show how one may simultaneously democratize strength, via scaling of γ, and the characteristic time or dispersion via scaling of η.

Democratization of Strength
Our immediate goal is to determine the dependence of the somatic strength, V 0 (0), on the location, amplitude and duration of the synapse.
The following discussion, summarized in Proposition 1, shows how to vary the amplitude g σ with the distance x σ of the synapse from the soma, so that the resulting impact, as measured by V 0 , is independent of x σ . We assume that membrane constants, as well as the duration of synaptic activation, as measured by T and its nondimensional form η, do not vary.
Integrating both sides of (3) over time gives while integrating (5) reveals Since solutions of (14) must decay to zero away from x σ it follows that and so, recalling Eq. (13), Here Q is obtained by integrating (8) up to time T . More precisely, where, with m ≡ γ 2 − 1, and This exact solution can now be used to choose g σ (x σ ) to render the somatic strength, V 0 (0), independent of synapse location, x σ . From (15)-(16) we find that this will follow if γ satisfies where V 0 is the desired, fixed somatic strength. On observing that Q(0, η) = 0, that γ → Q(γ, η) is increasing and defining we arrive at Proposition 1. Assume that the reversal potential, E, and the space and time constants (λ and τ respectively) are fixed. If the desired somatic strength, V 0 , effective synaptic duration, η = T /τ , and distance to the soma, x σ , together obey then there exists a unique synaptic conductance, g σ (x σ ), such that the associated somatic strength is indeed V 0 .
In Fig. 2 we present this equistrength synaptic conductance, g σ (x σ ), as a function of location, for a model cell for which We present results for different, but fixed, fiber radii a, that span a natural range. We note in each case that g σ exhibits moderate linear growth (less than 1 nS/100 µm) when the synapse is within 500 µm of the soma. Choosing from the middle of the range, e.g., a = 1 µm and g σ = 5 nS, corresponds to γ ≈ 0.19. We plot the associated synaptic and somatic potentials in Fig. 3 for a = 2 µm. We see that in order to achieve a strength of 10 mVms at the soma the synaptic potentials peak at less than 6 mV, even at 450 µm. This democratization of somatic strength is achieved by a simultaneous decrease in somatic peak and increase in somatic width.
Although (20) is exact, it does not yield a solution in closed form. Toward such an end we note that the right side of (20) is typically small, and so, for small γ, we may exploit distance from soma (µm) Figure 2. A: A plot of Q(γ, η) as a function of the nondimensional synaptic conductance γ (recall (16) and (17)) over modest γ. Results are presented for different values of the nondimensional synaptic duration η ranging from 0.2 to 3 in steps of 0.4. B: The associated equistrength synaptic conductance, xσ → gσ(xσ), achieved by solving Eq. (20) for the cell described in (22)-(23) with the specified somatic strength V 0 = 10 mVms and η = 1 and for several fiber radii: a = 2 µm (asterisks), 1 µm (triangles) and 0.5 µm (circles). The associated dashed curves offer the approximations, final expression in Eq. (24), corresponding to the synaptic model, Eq.
(2), of direct current injection. We now show that this latter value is precisely the 'equistrength' synaptic conductance that holds in the simplified setting of Eq.
(2) where the synaptic driving force is merely E. In this case the associated strength and so the associated equistrength conductance is U 0 exp(x σ /λ)/(ET ), as above. In Fig. 2 we contrast this conductance with the 'true' value obtained by numerically solving the implicit equation (20).  We next examine the limit of dendritic democratization of strength. The major difference between models (1) and (2) is that in the first case v(x σ , t) < E, regardless of g σ , while in the second case the voltage, u(x σ , t), may be unbounded. In the limit g σ → ∞, the voltage v(x σ , t) is clamped to E for t ∈ [0, T ] and decays exponentially afterwards. By monotonicity, the value of V * 0 (0) corresponding to this profile cannot be exceeded by any other conductance change which is nonzero only in the interval [0, T ]. Similarly, there is no value g σ that can result in a somatic strength V 0 that exceeds this value V * 0 (0). Given a desired strength, we can use the results of this section to compute the location beyond which democratization fails. By Proposition 1, a prescribed somatic strength V 0 can be achieved only for synapses located at a distance from the soma. In Fig. 4 we show how this distance scales with radius, a, over a large range of values of intracellular resistivity, R i (Golding et al., 2005).
We emphasize that for model (2), associated with direct current injection, no such restriction on the achievable somatic strength exists. In this case, there is no mathematical limit to dendritic democracy. There are certainly physiological limits associated both with the number of postsynaptic receptors per synapse and, more importantly, the conductance beyond which the synaptic potential will activate sodium and/or potassium currents and so contradict the assumption of a passive cable.

Democratization of Characteristic Time
We have seen that democratization of strength via increasing synaptic amplitude, g σ , alone leads to shorter but wider somatic responses. We now show how to scale the duration of the synaptic time course, T , in order to counteract this broadening. We accomplish this by focusing on two natural combinations of the first few moments of t → v(x, t).
With the synapse at x σ we first derive a simple law for transporting the synaptic characteristic time, C(x σ ), to the somatic characteristic time, C(0). We then show how to scale the duration of the synaptic input T in order that C(0) not vary with x σ , assuming that g σ is fixed. We end the section by showing that g σ and T can be scaled concurrently so that both C(0) and V 0 (0) are independent of x σ .
On taking the first moment of the two piece cable equation, (3) and (5), we find that V 1 (x) satisfies Therefore, /2 which, when reconciled with the above, yields, It follows that the characteristic time obeys the very simple transport law In order to democratize C we ask that the somatic characteristic time, C(0), take a prescribed value, call it C, independent of x σ . This compels us to solve C = C(x σ ) + τ x σ /(2λ) and so we proceed to compute where Note that Eqs. (18) and (19) provide expressions for the integrals occurring in the denominator of R(γ, η). Regarding the numerator, we record It follows that η → R(γ, η) is increasing and R(γ, 0) = 1/2. As a consequence, we obtain a counterpart to Proposition 1 Proposition 2. If γ is fixed and 1 + x σ /λ < 2C/τ then there exists a unique T = T (x σ , C) such that the somatic characteristic time is C.
We next turn to the simultaneous democratization of strength, V 0 (0) and characteristic time, C(0). The discussion in the present and previous section shows that this is equivalent to solving the following equations simultaneously for γ and η We contrast the exact solution (via fsolve in Matlab) with an explicit approximate solution that stems from the observation that η → R(γ, η) is very close to linear and very insensitive to γ. More precisely, R(γ, η) ≈ (1 + η)/2 permits us to solve Eq. (29) for which we may then place in our approximate solution, Eq. (24), of Eq. (28), to yield Fig. 5 illustrates that these approximations faithfully follow the true solution.
We now tie this approximation back to the characteristic time of the potential, u, of the cable, Eq. (2), with direct current injection. First note that our approximation of R when placed in Eq. (27) produces Now, as in Eq. (26), we find that the first moment of t → u(x, t) at x σ and so the associated synaptic characteristic time indeed coincides with Eq. (32).
We have plotted in Fig. 6 the synaptic and somatic potentials associated with the true simultaneous democratizers, with a = 2 µm, and note that the addition of characteristic time has indeed reversed the trend noted in Fig. 3. That is, the peak somatic response now increases as a function of the distance of the synapse from the soma. Of course, this comes at the price of large, fast distal input and associated large synaptic potentials (see Fig. 6A). This characteristic time is not the only measure of somatic timing. In fact we shall find it useful to contrast the findings of this section with results of simultaneous democratization of strength and dispersion.
As in the previous section, strength, V 0 (0) and dispersion, D(0) can be democratized simultaneously by solving the pair of equations We contrast the exact solution (via fsolve in Matlab) with an explicit approximate solution that stems from the observation that η → S(γ, η) is very close to quadratic and very insensitive to γ. More precisely, S(γ, η) ≈ 1/2 + η 2 /12, which permits us to solve Eq. (36) for which we may then place in our approximate solution, Eq. (24), of Eq. (35), to yield As above, these approximations faithfully follow the true solution (see Fig. 7). Moreover, the approximation jibes with the dispersion of the potential, u, of the cable, Eq.
(2), with direct current injection. First note that our approximation of S when placed in Eq. (34) produces Now, as in Eq. (33), we find that the second moment of t → u(x, t) at x σ satisfies and so the associated synaptic dispersion  Fig. 7 with a = 2 µm. We note in panel B that the somatic potentials become taller and thinner with increasing distance.
indeed coincides with Eq. (39). We have plotted in Fig. 8 the synaptic and somatic potentials associated with the true simultaneous democratizers, with a = 2 µm, of strength and diffusion. In comparison with the associated plots for the democratization of strength and characteristic time we note that the synaptic peaks and the variation in somatic peak with distance are both diminished. In addition, on comparing Fig. 5 to Fig. 7, we see that democratization of dispersion requires less drastic growth in g σ and decay in T with distance.
Despite the above, in all instances of "democracy" we see that g σ scales linearly with x σ for small x σ . This is consistent with the scaling laws found from experiments (Migliore et al., 2005). With increasing x σ we find that the democratic choice of g σ increases faster than linear.
Moreover, in all cases beyond some critical distance from the soma there is no choice of g σ that can achieve democracy. We could, of course, proceed to simultaneously democratize strength and sensible combinations, e.g., skew and kurtosis, of higher and higher moments in the hopes of getting finer and finer control on the variation in peak somatic amplitude. However, we leave such an approach for future study, and instead embark on a more direct attack on the peak amplitude.

Solution via a Neumann Series Expansion
As we have seen in §2 cable theory shows that the potential induced by a synaptic current depends nonlinearly on the conductance change. As for the case of a constant current injection one can define a Green's function, though now it is no longer time-translation invariant, that can be used to find the cable voltage. However, for practical calculations it is more useful to formulate the cable response in terms of a Dyson equation. In this way we can express the response to synaptic current in terms of the bare response function of the model without shunting. For a further discussion of this useful relationship we refer the reader to the review by Bressloff and Coombes (1997).
We begin by recasting the cable equation, Eq. (1), as an integral equation where G(x, t) is the Green's function of the infinite uniform passive cable expressed in Eq. (9). The implicit form of Eq. (41) suggests a Neumann series solution that can be obtained by repeated substitution of (41) into itself (Bressloff and Coombes, 1997;Timofeeva et al., 2006b). Performing a substitution just once, we have that We recognize the first term as the exact solution, u, Eq. (10), to the problem of direct current injection. After an additional iteration we (42) The integral term at second order can be obtained in closed form (via a Laplace transform) as

Hence, the solution of Eq. (1) that includes only the first two terms in the Neumann series expansion is
We note that, although Tuckwell (1988b) has found an exact solution for t < T , the use of (43) is much preferred since it can be used to find the maximum of v which always occurs at some t ≥ T . It is useful to make a comparison between cable responses to synaptic input using numerical solutions to (1) (equivalent to a non-truncated Neumann series solution) and the approximate solutions v N,2 (x, t) and u(x, t). In Fig. 9 we plot the membrane voltage in the cable at the location of the synaptic input. The three curves in each plot correspond to the numerical solution of Eq. (1) (dashed black curve), the approximate solution v N,2 (x σ , t) defined by Eq. (43) (red curve) and the solution u(x σ , t) given by Eq. (10) (blue curve). The numerical solution of the model was obtained using neuron (Carnevale and Hines, 2006). The model parameters are as in (22)-(23) with the cable radius, a equal to 2 µm in panel A, 1 µm in panel B, and 0.5 µm in panel C.
These plots nicely demonstrate that, for physiologically realistic parameter values, the approximate solution v N,2 (x σ , t) is in very good agreement with the numerical solution of the model for relatively large values of the radius (Figs. 9A and B). However, this is not the case when the radius of the cable is very small (Fig. 9C). Therefore for small radii more terms have to be included in the Neumann series expansion to improve accuracy. As expected, the solution u(x σ , t) of Eq. (2) offers a worse approximation than v N,2 (x σ , t).

Current injection versus conductance change
As shown above, the solution of the model with shunting synaptic input (1) can be written as a Neumann series in the nondimensional synaptic conductance γ. If γ ≪ 1, the contributions of the second and higher order terms in the expansion become negligible and the approximate solution is given by just the first term in (42). In this case the model with shunts will behave like the model without shunts given by Eq. (2).
For larger values of the parameter γ higher order terms in the Neumann series expansion cannot be ignored. As γ increases the second order correction becomes more important, up to the point where γ is so large that the Neumann series approximation v N,2 (x, t) breaks down.
For fixed values of R i and G L , γ is proportional to the ratio g σ /(2a) 3/2 . Clearly, this dependence of γ on the radius of the cable explains the results of Fig. 9 where both the full and approximate solutions are plotted for different values of a (showing worse agreement with decreasing a). To explore the parameter range in which the Neumann series approximations v N,i are accurate we calculate the L 2 norms of their difference from the full solution of Eq. (1). In particular, we plot t)] 2 dt) 1/2 for i = 1 and 2 in Fig. 10 as functions of γ with fixed g σ . The blue and red curves correspond to the case i = 1 and i = 2, and v N,1 (x σ , t) = u(x σ , t) and v N,2 (x σ , t) respectively. The inner plot in this figure is a magnified view of the area marked by a dashed rectangle, and demonstrates that the error between the full (numerical) and approximate solutions can be significantly reduced (for relatively large radii) by adding the second term in the Neumann series expansion. However, if the radius of the cable is too small (1/(2a) 3/2 > 2.5, i.e. a < 0.272) solution v N,2 is actually worse than v N,1 , and higher order terms in the Neumann series expansion are necessary. Parameters as in Fig. 9. Inner plot is a magnified view of the area marked by a dashed rectangle.
In Fig. 11 we use the data of Fig. 10 (for fixed g σ ) to quantify the parameter region in which the second order expansion is accurate (by demanding the solution satisfy |v| L 2 < 0.85). For example with (g σ , a) = (5, 2.5) we see that using just one term in the Neumann series expansion will give a poor approximation of the full solution, but that going to second order will give a good approximation.  Figure 11. Regions of the parameter plane (gσ, a) in which the L 2 norm of the difference between the full solution and the approximate solution, |v| L 2 , is bigger (less) than 0.85, describing a bad (good) approximation. The gray area in A (B) corresponds to vN,2 (u) providing a bad approximation. Hence, using the Neumann series expansion, the region in the (gσ, a) plane supporting a good approximation to the full solution increases with the inclusion of higher order terms, as expected.

Democracy: equalizing somatic response
In this section we use the second order Neumann series approximation to determine the scaling of the synaptic strength g σ and duration T necessary for "democratization." The impact of a synaptic input is characterized by the maximum of the depolarization at the soma, unlike in §3 where moment based measures were used. We first find conditions on g σ such that the maximum depolarization is independent of changes in the location of the synaptic input. We next find conditions such that, in addition, the maximum also occurs at the same point in time, regardless of the location of the input.

Equalizing voltage peak
We first consider the problem of equalizing the maximal depolarization at the soma. The only parameter that is allowed to vary is g σ = g σ (x σ ), the location-dependent synaptic strength. Our aim is to find a function g σ that will equalize the voltage peak at the soma located at x = 0, i.e.
we require max t>0 {v N,2 (0, t)} = V for any x σ . The time t * at which the maximum voltage is reached satisfies t * ≥ T . This will depend on the location x σ and can be found by solving ∂v N,2 (0, t)/∂t = 0, where v N,2 (0, t) is given by (43). The time t * for each location x σ satisfies v N,2 (0, t * ) = V . Thus, the scaling law for g σ = g σ (x σ ) can be found by simultaneously solving the following system of equations for the pair (g σ , t * ). Figure 12A shows how the synaptic strength has to scale along the cable to equalize the maximum response at the soma.
Since T is fixed, the times t * at which these maximal responses occur increase with x σ . An example of this is shown in Fig. 12B. In Fig. 13  Parameters as in Fig. 9 with V = 2 mV and a = 2 µm (asterisks), a = 1 µm (triangles), a = 0.5 µm (circles).
we plot the synaptic and associated somatic potentials (Fig. 13A) and three examples of voltage profiles at the soma for different synaptic contact points (Fig. 13B). We note that the curves in Figs. 12 terminate at finite values of x σ because a solution to the pair of equations (44) and (45) ceases to exist. In parameter regimes where v N,2 is a poor approximation to the full solution this termination distance is an underestimate of the actual distance over which democracy can be guaranteed.  Figure 13. A: Synaptic and associated somatic potentials for a synaptic input at xσ. B: Associated somatic potentials when the synaptic contact is at the distances of 90, 290 and 490 µm from the soma. Parameters as in Fig. 12 with a = 2 µm.

Equalizing voltage peak and its time-to-peak
We next examine the conditions under which the amplitude and timeto-peak at the soma is independent of the location of the synaptic input (t * fixed). To accomplish this we let the duration of the synaptic input T vary with the location of the input and write T = T (x σ ). This more constrained form of "democracy" can be analyzed by solving (44) and (45) for (g σ , T ). Results of such a calculation are shown in Fig. 14. In Fig. 15 we show the synaptic and associated somatic potentials (Fig. 15A) and three examples of voltage profiles at the soma for different synaptic contact points (Fig.15B).  Figure 14. A plot of the functions gσ and T (A and B) that equalize the maximum voltage and the time-to-peak at the soma. Here t * = 3.35 ms, V = 2 mV, a = 2 µm (asterisks), a = 1 µm (triangles), a = 0.5 µm (circles), and other parameters as in Fig. 9.
As in §3, we see that "dendritic democracy" is ensured for g σ that scales linearly close to the soma and faster than linearly further away. This is seen in Figs. 12A and 14A, where it can also be seen that the critical distance beyond which democracy cannot be achieved decreases with decreasing cable radius. One may ask whether these results will hold for more complicated dendritic geometries relevant to real neurons. In the next section we introduce the appropriate mathematical techniques to answer this question and show that the answer is indeed yes.

Branched dendritic structures
Let us assume that we have an arbitrary branching dendritic structure and that the voltage dynamics v i (x, t) on each branch (with local spatial coordinate x), labeled by i, is given by a passive cable equation. The synaptic conductance on branch j at location x j is given as g j (t) = g σ H(T − t). Then assuming v i (x, 0) = 0 for all branches, the voltage on each branch can be found as where G ij (x, x j , t) is the Green's function of the given branching structure. This can be constructed using the "sum-over-trips" framework (Abbott et al., 1991), recently extended to tackle the inclusion of the soma and allowing for variation of parameters across branches (Coombes et al., 2007). Formally speaking if we denote the Green's function of an infinite cable by G (Eq. (9)) then where a "trip" depends on the labels (i, j, x, y) and there are an infinite number of trips and trip coefficients A trip . We refer the reader to (Abbott et al., 1991;Coombes et al., 2007) for complete details. Following our earlier approach for an infinite cable we again pursue a Neumann series expansion. The first two terms in the series are The first term can be calculated using an infinite sum as g σ r j E/(2λ j ) trips A trip K(L trip (i, j, x, y), t), where K(x, t) = A(x, t − t∧T )−A(x, t) and A(x, t) is given by (11). The second term is obtained from G ij (x, y, t) by numerical integration. We now consider an example of a real neural geometry as shown in Fig. 16 and determine how the synaptic conductance scales along the apical trunk (shown in blue) to achieve (voltage) democracy. Further, we make a comparison between the full solution (obtained using neuron) for the case where the synapse is modeled as in Eq. (1), the second order approximate solution given by (47), and the solution for the case where the synapse is modeled as in Eq. (2), so that the solution is simply the first term in (47). In Fig. 17 we show the democracy plots for the realistic neuron. It is apparent that the first two terms of the Neumann series capture the behavior of the full model (obtained numerically), and that the trend in scaling is the same as for the unbranched case (cf. Fig. 12A). (nS) Figure 17. A: gσ as the function of distance (along the blue path in Fig. 16) that equalizes the maximum voltage V = 2 mV at the soma. Asterisks: first order approximation, circles: full numerical solution, triangles: second order approximation. Inner plot is a magnified view of the area marked by a dashed rectangle. Biophysical parameters across the tree are as given in (22)-(23), radii vary, the synaptic duration T = τ . B: Voltage responses in the soma.

Discussion
In this paper we have focused on several key measures of "dendritic democracy" using two complementary approaches -one relying on moment methods and the other on a Neumann series solution. When combined, these two approaches suggest that the synaptic conductance has to scale linearly close to the soma and faster than linearly further away. Moreover, beyond some critical distance there is no choice of conductance that will ensure democracy in the presence of shunts. This suggests that, for distal synaptic inputs, democracy cannot be achieved by simply increasing synaptic conductance strength and that other mechanisms have to be invoked. For example, the attenuation of voltage as it propagates from the distal dendrites to the soma might be compensated for by active currents. Indeed, dendritic P-type Ca 2+ channels have already been studied as possible mediators of synaptic amplification via dendritic spikes to ensure democracy in a computational model of a cerebellar Purkinje cell (De Schutter and Bower, 1994). Clearly, the inclusion of nonlinear membrane dynamics limits any quantitative mathematical analysis. However, recent work on the spike-diffuse-spike model has shown that this provides a reasonable caricature of a tree with active conductances and, importantly, is mathematically tractable (Timofeeva et al., 2006a;Timofeeva et al., 2006b). Our approach for studying "dendritic democracy" can be treated within this framework and will allow a systematic exploration of the amplifying nature of active spines on distal synaptic inputs. Moreover, it is believed that somatic response to distal inputs can be boosted not only by supra-threshold, but also by subthreshold (resonant) dynamics, for example via I h and I A channels. The mathematical approach of this paper can be further extended to cover resonant branched dendrites using techniques developed in (Coombes et al., 2007). Finally, we note recent work showing that the tapering of real dendrites can equalize the current transfer from all synaptic locations (Cuntz et al., 2007). It would be interesting and worthwhile to extend the type of analysis presented here to this case.