The Endurance and Selectivity of Spatial Patterns of Long-Term Potentiation/Depression in Dendrites under Homeostatic Synaptic Plasticity

The residual potentiation after HSP in an isopotential neuron model

Long-term potentiation (and diametrically, long-term depression) of excitatory synapses is expected to increase the average activity of the postsynaptic neuron. If persisting for a long period of time, this increased activity will be counterbalanced by regulatory mechanisms such as HSP that will decrease synaptic strength, so that the activity of the neuron can resume its target level. Therefore, after HSP, at least some of the initial synaptic potentiation, p⁰, is expected to be lost, leaving what we shall call a residual potentiation, rp, the net increase in the original synaptic strength resulting from the joint effects of LTP and HSP. What will be the size of the residual potentiation of a synapse after HSP (in what follows, we mostly refer to synaptic potentiation, but our conclusions symmetrically hold for synaptic depression)?

Several insights on this question may be gained by contemplating a very simple model. Consider a dendritic branch receiving N synaptic inputs. On the one extreme, this branch could be divided into N electrically decoupled isolated compartments such as illustrated in Figure 1A (top). On the other extreme, this branch may be effectively isopotential (Fig. 1A, middle). Normally, the cable properties of dendritic branches lie somewhere between these two extremes, so that there will be current flow (or, e.g., a gradient of calcium concentration) along the dendritic cable, and membrane potential will vary between different synaptic locations (Fig. 1A, bottom). With a few simplifying assumptions, the amount of residual potentiation, rp, can be readily analyzed for the two extreme cases: complete electrical isolation and the isopotential case. At the heart of this analysis is a simple model for the HSP process that describes the very slow change of the peak synaptic conductance, G_syn,i of each synapse, i, according to the difference between membrane potential, V, and a target membrane potential, V_trg: where τ_H is the time constant for the HSP process, which is very large relative to the membrane time constant τ_m, and κ = 1, in voltage units, maintains G_syn,i in units of conductance. The solution to this equation for the isolated case (Fig. 1A, top) (supplemental Appendix I, Equation A7, available at www.jneurosci.org as supplemental material) is shown in Figure 1B, where rp(T), the amount of residual potentiation as a function of time (in units of τ_H; T = t/τ_H) is plotted for initial potentiations p⁰ = 150 and 200%, as well as for a 50% initial depression of a synapse on the branch. Obviously, in this case, any potentiation is bound to decay to nil at the HSP timescale (of hours to days), eventually leaving no residual potentiation.

Conversely, in the case of an isopotential dendrite (Fig. 1A, middle), the HSP dynamics will multiplicatively scale down all synaptic weights by a global scaling factor, SF (supplemental Appendix I, available at www.jneurosci.org as supplemental material). As a consequence, in addition to the potentiated synapses, the other nonpotentiated synapses will also be weakened, thus supporting the potentiated synapses so that the latter will not necessarily lose all of their initial potentiation. How much of the initial potentiation will be rescued is a function of the number of potentiated synapses, n_p, relative to the total number of synapses, N. As an example, we have derived this function for the case in which all n_p synapses are initially potentiated to the same level, p⁰ (supplemental Appendix I, Equation A10, available at www.jneurosci.org as supplemental material). This is plotted in Figure 1C (for p⁰ = 150, 200, and 50%, as in Fig. 1B). In the case in which the number of potentiated synapses is very small relative to the total number of synapses (n_p/N → 0 in Fig. 1C), these synapses will maintain their initial potentiations almost fully (rp will approach p⁰). In contrast, as the number of potentiated synapses approaches the total number of synapses (n_p/N → 1 in Fig. 1C), the potentiation will be entirely lost (rp will approach 100%). In fact, this is what happens in the isolated case where n_p = N = 1.

HSP in a detailed neuron model

What would be the residual potentiation of synapses in a more realistic, electrically distributed dendritic tree? It can be expected to lie somewhere between the isopotential residual potentiation (Fig. 1C) and the complete abolishment of the initial potentiation, in the isolated case (Fig. 1B). As in the isopotential case, nonpotentiated synapses can help support the persisting potentiation of synapses, but the location (spatial distribution) of the participating synapses along the dendrite will now become important. To investigate this under much more realistic conditions, we used a reconstructed model of a CA1 pyramidal neuron (Fig. 2D) with active channels in the axon and dendrites and multiple (587) excitatory synapses spaced at equal intervals of 20 μm throughout the dendrites (Fig. 2D, circles) (see Materials and Methods). The peak synaptic conductance, G_syn,i, of each synapse, i, was continuously modified according to the HSP mechanism described in Equation 1. The two possible variants of HSP were formulated. The first, g-HSP equally changed all synaptic weights according to the difference between the soma membrane potential, V_soma (a possible global measure of activity) and a target membrane potential, V_trg: Note that the conductance of all synapses changes according to the exact same expression on the right side of Equation 1. One could postulate a biological mechanism that can implement this, for example by changing the rate of relevant mRNA transcription at the cell nucleus with respect to somatic levels of activity.

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Figure 2.

The time course of HSP after a massive overall increase in activity in a detailed model of a CA1 pyramidal neuron. A, Average membrane potential, 〈V_i〉, at each synaptic site, i (circles) for a uniform random activation of all synapses (circles in D) at an average rate of 2 Hz before HSP (top profile) and after l-HSP (bottom profile). Horizontal dashed line marks the target membrane potential set at V_trg = −60 mV. B, Peak synaptic conductance, G_syn,i, for each synapse, i (circles) before HSP. All synapses had initially the same G_syn,i = 1 nS. C, G_syn,i distribution after l-HSP. The G_syn,i of a representative synapse (dark circle) is indicated by an open arrow. The time at which this G_syn,i distribution was obtained is marked as t₀ (see below). D, Reconstructed CA1 pyramidal neuron (kindly provided to us by N. Spruston). Circles mark the locations of 587 equally spaced (20 μm intervals) excitatory synapses on the dendrites. Schematic pipette indicates the representative synapse from C. E, F, G_syn,i distributions after HSP in response to an overall increase in activity produced by doubling the average synaptic activation rate from 2 to 4 Hz starting from time t₀ for g-HSP (E) and for l-HSP (F). The G_syn,i of the representative synapse is indicated in each plot by an arrow. In both cases, HSP scaled down G_syn,i values by half by the time marked as t_H. G, Time course of HSP downscaling of the G_syn,i of a representative synapse for g-HSP (light trace) and l-HSP (dark trace). Open arrow points at the G_syn,i value at time t₀ (as in C). Filled arrows point at the G_syn,i value at time t_H (corresponding to E and F). The traces are shown for an additional prolonged period of time, t₁. The inset presents the two traces for the time between t₀ and t_H, when most of the HSP scaling of G_syn,i took place.

The second, l-HSP controlled the local membrane potential, V_i, at each synaptic site, i (V_trg was assumed to be identical for all dendritic sites): In this scheme, each synapse changes differentially according to the local level of activity, which reflects how activity transpiring throughout the dendritic branch is integrated at the site of that synapse. Biologically, this could occur through local control of protein synthesis at the site of each synapse (Sutton et al., 2004, 2006).

With all synapses set initially at a uniform peak conductance of G_syn,i = 1 nS (Fig. 2B) and each synapse activated randomly and asynchronously at 2 Hz, the average membrane potential at each synaptic site, 〈V_i〉, assumed a very non-uniform shape (Fig. 2A, “prior to HSP”). We set V_trg = −60 mV (Fig. 2A, horizontal dotted line) and applied the l-HSP rule to all dendritic synapses. After l-HSP, 〈V_i〉 values all leveled at the V_trg line (Fig. 2A, “local HSP”) and the emergent distribution of synaptic weights became highly non-uniform (Fig. 2C), with a distance-dependent increase in synaptic strength along the initial 500 μm of the main apical trunk, reminiscent of that found in CA1 pyramidal neurons (Magee and Cook, 2000). We demonstrated previously (Rabinowitch and Segev, 2006) that g-HSP, in contrast to l-HSP, will proportionally scale up or down the initial distribution of synaptic weights and will not necessarily level all 〈V_i〉 values to the desired V_trg level. This also occurred in the example presented in Figure 2 (data not shown).

To appreciate the model HSP timescale, we ran a simulation similar to the typical experimental paradigm used to study HSP, in which the activity of a neuron is globally enhanced, e.g., by pharmacologically reducing inhibitory synaptic input to the neuron (Lissin et al., 1998; O'Brien et al., 1998; Turrigiano et al., 1998). In the model, this was implemented by doubling the average frequency of synaptic activation from 2 to 4 Hz, starting with the initial spatial distribution of synaptic weights as in Figure 2C, in which 〈V_i〉 = V_trg. g-HSP and l-HSP responded to such overall perturbation similarly (Rabinowitch and Segev, 2006), by uniformly scaling down all synaptic weights (Fig. 2E,F). The time course of the change in conductance of a representative synapse (indicated in Fig. 2D by the schematic pipette and by arrows in Fig. 2C,E,F) is plotted in Figure 2G. As can be seen, most of the downscaling was completed for both l-HSP and g-HSP by time t = t_H (Fig. 2G, inset). We continued the simulation for a long time beyond t_H, until t = t₁. During this time, there was not much additional change in G_syn,i. Thus, under conditions of overall perturbation of activity, the time between t₀ and t_H can be considered to be equivalent to the many hours to days time course of HSP under comparable experimental conditions.

Residual potentiation of a single potentiated dendritic synapse

Research on LTP has contributed an immensely rich diversity of both theoretical and experimentally founded rules for the induction of LTP (Sjostrom et al., 2001; Abarbanel et al., 2003; Froemke et al., 2005). However, these rules vary greatly, and, in addition, the dependence of LTP induction on dendritic location is only beginning to be understood (Froemke et al., 2005). We therefore decided not to implement a specific activity-dependent LTP rule in our model but instead to assume that certain synapses underwent potentiation and to artificially increase the strength (peak conductance) of these potentiated synapses by some initial value, p⁰. This has the additional benefit of rendering our conclusions independent of the exact mechanism governing LTP induction.

Figure 3, A and B, shows an initial potentiation of the representative synapse from Figure 2C to p⁰ = 150% at time t = t₀ (Fig. 3A,B, potentiated synapse in red and immediate two neighboring synapses in blue). As in Figure 2C (and unless otherwise mentioned), all synapses were activated randomly at 2 Hz. The potentiation of this single synapse (of the large number of synapses occupying the dendrites), located 330 μm from the soma, went unnoticed by g-HSP (data not shown) and will thus persist indefinitely. In contrast, l-HSP responded by weakening the initial potentiation of the synapse to a residual potentiation of ∼140% (Fig. 3C, red). In addition, it also slightly weakened the nearby nonpotentiated synapses (Fig. 3C, blue). This “lateral suppression” created a new balance, allowing the potentiated synapse to maintain its potentiation and thus supporting it. The time course for the drop in residual potentiation is plotted in Figure 3D (solid red trace). Also plotted, for comparison, is the decay in residual potentiation in the isolated case (dashed red trace, reproduced from Fig. 1B) and the suppression of the two neighboring synapses (blue overlapping traces). Recall that t = t₁ is equivalent to a very long time period in real time (compare with Fig. 2G). Thus, we find that the potentiation of a single synapse can persist even in the presence of an l-HSP mechanism, thanks to the slight weakening of its nearby nonpotentiated neighboring synapses. How does such supporting lateral suppression come about? This predicted phenomenon is attributable to the distance-dependent interactions occurring along the dendritic branch. The potentiation of a synapse i (Fig. 3B, red synapse) will naturally result in an increase in 〈V_i〉 at the site of that synapse but also in an increase in 〈V_i〉 in neighboring sites j (Fig. 3B, blue synapses), causing l-HSP to weaken these neighboring synapses together with synapse i. The weakening of the neighboring synapses will, in turn, contribute to driving 〈V_i〉 values back to V_trg, forming a new stable constellation of synaptic weights as in Figure 3C.

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Figure 3.

Residual potentiation of a single potentiated dendritic synapse after HSP. A, The initial G_syn,i distribution at time t₀, similar to Figure 2C, except for a 150% potentiation of G_syn,i of the representative synapse (red circle and open arrow). The two immediate neighbors of the potentiated synapse are marked in blue. B, The initial potentiation at time t₀, obtained by dividing all the G_syn,i values in A by the corresponding values before potentiation in Figure 2C. C, The residual potentiation after l-HSP at time t₁, obtained by dividing all G_syn,i values at time t₁ in the potentiated condition by the corresponding values in a control condition. D, Time course of the residual potentiation of the representative potentiated synapse (red) and of the suppression of its two immediate neighboring synapses (blue). Also plotted, for comparison, is the decay in the initial potentiation of a single synapse in the hypothetical case of an isolated compartment (dashed red trace, as in Fig. 1B). E, The residual potentiation after postsynaptic l-HSP at time t₁, for a presynaptic 150% potentiation of the representative synapse (average activation rate of that synapse was raised from 2 to 3 Hz).

The expression of LTP could also occur presynaptically (Reid et al., 2004). In Figure 3E, we demonstrate that the same principle shown in Figure 3C also holds for this case. We “potentiated” the representative synapse presynaptically by increasing its average activation rate by 150% (from 2 to 3 Hz). This could be interpreted, for example, as an increase in release probability. In principle, the complete abolishment of this potentiation by a postsynaptic HSP mechanism would be the reduction of the peak synaptic conductance to 1/1.5 = 66.7% of its original strength. l-HSP indeed weakened the conductance of the potentiated synapse, but only to 90% (Fig. 3E, red). Once again, lateral suppression hindered any additional loss of the potentiation of the synapse.

Residual potentiation of a group of potentiated dendritic synapses

What happens in an electrically distributed dendrite when more than just one synapse is potentiated? As we have seen in the isopotential case, the number of potentiated synapses, n_p, compared with the total number of synapses, N, is of key importance in determining the residual potentiation. Is this also true for the realistic neuron?

In Figure 4, we present the residual potentiation after l-HSP of the representative synapse from Figure 3 under three different conditions: (1) the reference case in which only that synapse had been potentiated (Fig. 4A, left, B, red trace as in Fig. 3D); (2) a total of n_p = 5 adjacent synapses, including the representative synapse (Fig. 4A, middle), were all potentiated to p⁰ = 150%; this increase in n_p resulted in a big reduction in the residual potentiation of the synapse (Fig. 4B, green trace); and (3) the same number (n_p = 5) of synapses was potentiated, but this time the synapses that underwent potentiation were more spread out (Fig. 4A, right); now the residual potentiation was larger (Fig. 4B, compare orange trace with green trace).

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Figure 4.

Residual potentiation after l-HSP depends on both the number of potentiated synapses and their spatial distribution. A, Illustration of the three tested conditions. Left, The reference condition as in Figure 3D, in which a single representative synapse is potentiated. Middle, A subgroup of n_p = 5 adjacent synapses including the representative synapse are potentiated. Right, A subgroup of n_p = 5 spread-out synapses including the representative synapse are potentiated. B, Residual potentiation time courses for the three conditions in A. The colors of the traces correspond to the colors of the schematic pipettes in A. C, The SIE of the representative synapse plotted for a transient synchronized and enhanced (10 Hz) input to the clustered (green) or disperse (orange) subgroups of n_p = 5 synapses highlighted in A (middle and right, respectively). SIE is plotted at three time points: before any potentiation has occurred (“before potentiation”) with all synaptic weights (including those of the subgroup synapses) equal to their steady-state values as in Figure 2C; after a p⁰ = 150% potentiation of the subgroup synapses at time t₀ before HSP (“t₀ before HSP” as in B); and at time t₁ after l-HSP (“t₁ following HSP” as in B). D, The residual SIE at times t₀ and t₁ derived by dividing the SIE values for each condition by the corresponding SIE values before potentiation.

Although consisting of the same number of synapses (n_p = 5), the “clustered” (Fig. 4A, middle) and “dispersed” (Fig. 4A, right) patterns are distinguishable at the level of the dendritic branch and were thus subject to different levels of suppression by l-HSP (Fig. 4B, green and orange traces). Are these local variations in spatial distribution also detectable by the soma? What is their impact on the output of the postsynaptic neuron?

To address these questions, we designed a simulation in which the input from the subgroup of the five potentiated synapses transiently changes to become synchronized and its activation rate is increased from 2 to 10 Hz (the rest of the synapses remain at 2 Hz). This momentary change could represent some sort of event that the postsynaptic neuron is expected to detect, occurring at the short timescale (too brief for HSP to respond to). To assess the impact of the representative synapse on the firing of the neuron within different spatial patterns, we used the SIE measure (London et al., 2002), which estimates the mutual information (in bits per second) between the presynaptic and postsynaptic firing patterns (see Materials and Methods). First, we compared between the SIEs of the clustered (green) and dispersed (orange) patterns before any synaptic potentiation (Fig. 4C, “before potentiation”). In this case, the representative synapse and its four adjacent neighbors (Fig. 4A, middle) or four scattered neighbors (Fig. 4A, right) received the enhanced synchronized input with all synaptic weights as in Figure 2C. Even before any potentiation was introduced, the two spatial patterns were discernable at the soma as measured by the corresponding SIEs. The representative synapse was 1.75 more effective in influencing the firing of the postsynaptic neuron when participating in the dispersed subgroup than in the clustered one (compare between the light orange and green columns in Fig. 4C, “before potentiation”). The smaller synaptic efficacy in the clustered case can be explained by the reduction in the driving force and local input resistance of the synapses attributable to the spatial proximity of the synapses.

The p⁰ = 150% potentiation of the synapses in the subgroup substantially enhanced the SIE in both the clustered (green) and dispersed (orange) conditions (Fig. 4C, “t₀ prior to HSP”). The 150% potentiation of the five synapses resulted in an ∼150% increase in the SIE in the clustered case (compare leftmost and middle green columns in Fig. 4C whose ratio is shown in Fig. 4D, green column, “t₀ before HSP”) and an ∼200% increase for the dispersed case (Fig. 4D, orange column, “t₀ before HSP”). After l-HSP, which slowly weakened the weights of the potentiated synapses (Fig. 4B), the SIE also decreased for both patterns (Fig. 4C, “t₁ following HSP”). However, in the clustered case, the SIE dropped to its value before potentiation (Fig. 4D, green column, “t₁ following HSP”), whereas in the dispersed pattern, there remained a substantial amount of residual SIE (133%) after HSP (Fig. 4D, orange column, “t₁ following HSP”).

Thus, in the realistic neuron, under l-HSP, it is not only the number of potentiated synapses, n_p, that matters but also their spatial distribution. At the large timescale, l-HSP will favor synaptic potentiations that occur within a sparse spatial pattern of potentiations over a dense pattern. Coincidently, at the short timescale, a synapse pertaining to such a sparse pattern, when activated in synchrony with its partners, will have a stronger impact on the firing of the neuron. In contrast to l-HSP, g-HSP did not have the spatial resolution to distinguish between the two patterns in the above example (data not shown). An extensive comparison between l-HSP and g-HSP follows.

The survival of massive synaptic potentiations under HSP in dendrites

We further explored the effect of both l-HSP and g-HSP on varied spatial patterns of synaptic potentiations. In our model neuron, we selected the 50 longest terminal branches in the dendritic tree and substantially increased the number of synapses on these branches by decreasing the interval between them to 5 μm (increasing the total number of excitatory synapses on the dendrites to 1514) (see Materials and Methods). We ran a l-HSP simulation under these conditions to obtain the initial distribution of synaptic conductances (data not shown). For each branch, a probability for potentiation, pLTP, was drawn from a uniform distribution ranging from 0.1 to 0.9, and each synapse along the branch was potentiated with that probability to 150% of its initial strength. All in all, 591 of a total of 1514 synapses were potentiated (n_p/N > 1/3). This is illustrated in Figure 5A (top), in which the red filled circles represent the synapses that were potentiated (the red bars rising from them represent 150% potentiation). The blue filled circles represent the nonpotentiated synapses sharing the same branches, and the gray filled circles represent the rest of the synapses on other parts of the dendritic tree. Two representative branches are indicated by the open arrows: an oblique branch with pLTP = 0.12 had just two potentiated synapses (of 30) and a basal branch with pLTP = 0.85 had 18 potentiated synapses (of 20). The residual potentiation of these synapses after a long time, t = t₁, is presented for both l-HSP (Fig. 5A, middle) and g-HSP (Fig. 5A, bottom). We ran these simulations for an additional equally long period of time (until t = t₂) and compared between the residual potentiations at t₂ and t₁. They showed a clear match for both g-HSP and l-HSP (data not shown), indicating that these were steady-state residual potentiations.

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Figure 5.

HSP selecting between different spatial patterns of synaptic potentiation. A, Top, Synapses occupying the 50 longest dendritic terminals were randomly selected for potentiation according to a branch-specific probability for potentiation, pLTP. In these branches, synapses were densely spaced 5 μm from one another. The potentiated synapses are shown in red (p⁰ = 150% for all potentiated synapses, represented by the red bars). The neighboring nonpotentiated synapses are in blue. The rest of the synapses on the tree (spaced at 25 μm intervals) are in gray. Open arrows point at two representative branches, an oblique branch with pLTP = 0.12 and a basal branch with pLTP = 0.85. The amount of residual potentiation is represented by the size of the red bars, and the extent of suppression of the nonpotentiated synapses is represented by the size of the blue and gray downward pointing bars, after l-HSP (middle) and g-HSP (bottom). B, Residual potentiation levels of the two representative branches (from A; left, oblique branch with pLTP = 0.12; right, basal branch with pLTP = 0.85) for time t₀ (top, potentiated synapses aligned with top dotted red lines, labeled “t₀”); for time t₁ after l-HSP (middle); and after g-HSP (bottom, potentiated synapses aligned with the g-HSP dashed thick red lines, labeled “t₁ global”). Nonpotentiated synapses are also shown (blue and gray circles as in A). Middle dotted blue lines mark 100% of original synaptic strength, and bottom dashed thick blue lines mark the level of suppression after g-HSP.

The length of the red bars in Figure 5A is proportional to the residual potentiation, decreased in both cases attributable to the overall large number of potentiated synapses (n_p/N > 1/3). In addition, under both HSP variants, many neighboring synapses on the selected branches and throughout other parts of the dendrites also weakened (downward pointing blue and gray bars). However, although the level of residual potentiation was uniform for all synapses under g-HSP (132%) (Fig. 5B, bottom marked by thick red dashed lines), it was strongly branch specific under l-HSP. This can be appreciated for the two representative branches. The oblique branch with pLTP = 0.12 was only mildly affected by l-HSP (Fig. 5B, middle left). In contrast, the synapses on the basal branch with pLTP = 0.85, lost most of their initial potentiation (Fig. 5B, middle right). Thus, l-HSP, unlike g-HSP, will differentially and selectively affect the residual potentiations of synapses. Because l-HSP can operate at the level of functional dendritic compartments (Rabinowitch and Segev, 2006), it will be sensitive to the relative number of potentiated synapses (n_p,k/N_k) within each given compartment, k, whereas the g-HSP mechanism will respond to the total number of potentiated synapses in the entire tree (n_p/N).

Predicting the residual potentiation from the initial spatial pattern of synaptic potentiations

Can information on the remaining residual potentiation after HSP be extracted from the initial spatial pattern of synaptic potentiations? Figure 6A presents the residual potentiations of the synapses from Figure 5 as a function of their initial level of potentiation. Although all potentiated synapses shared the same level of initial potentiation, p⁰ = 150%, they exhibited a very wide range of residual potentiations under l-HSP (Fig. 6A, vertical red bar showing residual potentiation values ranging approximately between 100 and 150%) and a particular uniform level of residual potentiation of 132% under g-HSP (Fig. 6A, thick dashed red line). Similarly, the nonpotentiated synapses in Figure 5 began at 100% of their strength at time t₀ and were suppressed by l-HSP to levels ranging approximately between 70 and 100% of their initial strength (Fig. 6A, blue vertical bar) and to 88% of their initial strength by g-HSP (Fig. 6A, thick dashed blue line). Thus, the residual potentiation cannot be predicted from the initial potentiation levels alone. What additional information is needed?

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Figure 6.

Predicting the residual potentiation from the initial spatial pattern of synaptic potentiations. A, Residual potentiation at time t₁ from Figure 5, plotted against the initial potentiation at time t₀. The initial potentiation for all potentiated (red) synapses was 150%. The residual potentiation after l-HSP ranged between 100 and 150% (vertical red bar) and was 132% for all synapses after g-HSP (thick red dashed line). The remaining nonpotentiated (blue) synapses with 100% of their original strength at time t₀ were suppressed to 70–100% of their original strength after l-HSP (vertical blue bar) and to 88% of their original strength after g-HSP (thick blue dashed line). The diagonal is the equality line. B, Residual potentiation at time t₁ (with initial synaptic conductance as in Fig. 2B) after g-HSP plotted against the spatial dispersion index, SD_j, of the initial potentiation pattern (see supplemental Appendix II, Equation A11, available at www.jneurosci.org as supplemental material) for various extents of potentiations (n_p/N = 0.1, 0.25, and 0.5). Inset, Schematic examples of the three different spatial dispersion configurations used (bars represent n_p = 3 potentiated synapses). Top of inset, The extreme peripheral spatial pattern, j_max, corresponding to SD_jmax = 1 (cyan bars). Bottom of inset, The extreme somato-centric spatial pattern, j_min, corresponding to SD_jmin = 0 (red bars). Middle, An intermediate potentiation pattern, j with 0 < SD_j < 1 (normalized according to the above two extreme cases; turquoise bars). C, Residual potentiation at time t₁ (with initial synaptic conductance as in Fig. 2B) after g-HSP plotted against the relative number of potentiated synapses, n_p/N. Each curve represents a different spatial pattern of synaptic potentiations with varying spatial dispersions (SD = 1, 0.5, and 0, in cyan, turquoise, and red, respectively). Also plotted is the theoretical curve for the isopotential case (from Fig. 1C; dashed black) and results from a set of simulations with somato-centric (SD_i = 0) initial potentiation patterns in which action potentials were blocked (dashed red, “no AP”). D, Average residual potentiation (from Fig. 5) of the synapses at each branch at time t₁ after l-HSP plotted against the initial probability of potentiation of that branch, pLTP (each dot represents a branch) for initial potentiations of 150% (red) and 200% (green) and an initial depression of 50% (orange). Superimposed are the corresponding theoretical curves for the isopotential case, reproduced from Figure 1C. E, Residual potentiation (from Fig. 5) of each synapse at time t₁ after l-HSP plotted against the spatial segregation index, ss_i (supplemental Appendix II, Equation A12, available at www.jneurosci.org as supplemental material), for that synapse with α = 46 μm (see Results; each dot represents a synapse). Red and blue represent the potentiated and nonpotentiated synapses, respectively. Diagonal is the equality line.

Although under g-HSP all synaptic weights are modified in unison, this does not necessarily mean that all patterns of synaptic potentiations will be scaled by the same factor (as in the case of the isopotential neuron). In effect, in an electrically distributed neuron, the influence of synaptic potentiation on the global activity level [such as the value of V_soma or the average rate of action potential (AP) firing] is expected to vary according to the spatial distribution of these potentiations. For example, n_p potentiated synapses that are clustered close to the soma will have a much greater influence on V_soma than if the same number of potentiated synapses are located distally, at remote dendritic arbors. Therefore, even under a g-HSP mechanism (that is sensitive to the level of activity at the soma), the spatial pattern of the potentiated synapses is expected to determine the degree of their remaining residual potentiations. In Figure 6, B and C, we present quantitatively the influence of the extent of the spatial dispersion of a given pattern of synaptic potentiations on the remaining residual potentiation after g-HSP, using a measure for the degree of spatial dispersion, SD_j, for a specific spatial pattern of potentiations, j (supplemental Appendix II, Equation A11, available at www.jneurosci.org as supplemental material). SD_j may vary between 0, indicating no dispersion (i.e., all potentiated synapses are located as close as possible to the soma) (Fig. 6B, inset, bottom configuration), and 1, indicating maximal dispersion (i.e., all potentiated synapses are as distant as possible from the soma) (Fig. 6B, inset, top configuration). Intermediate values correspond to intermediate configurations (Fig. 6B, inset, middle configuration).

We ran several simulations using the model from Figure 2, with initial G_syn,i as in Figure 2B, for various spatial potentiation patterns with different spatial dispersion indices (SD_j = 0, 0.5, 1) and with different numbers of synapses initially potentiated to p⁰ = 150% (n_p/N = 0.1, 0.25, and 0.5). Figure 6, B and C, depicts the residual potentiations after g-HSP for these simulations. The residual potentiation under g-HSP was indeed larger with increased spatial dispersion (Fig. 6B). At the same time, an increase in n_p/N reduced the size of the residual potentiation. Thus, both the number of potentiated synapses and their spatial distribution throughout the entire dendritic tree will determine the size of the residual potentiation after g-HSP. In Figure 6C, we plotted the residual potentiation as a function of the number of potentiated synapses for each spatial dispersion pattern and compared these plots with the theoretical curve for the isopotential case (dotted black line; reproduced from Fig. 1C, for the p⁰ = 150% case). The extremely peripheral potentiation patterns (SD_j = 1; cyan filled circles) showed larger residual potentiations than the corresponding potentiations of the same number of synapses in an isopotential neuron (black filled circles). The extremely somato-centric potentiation patterns (SD_j = 0, red filled circles) had smaller residual potentiations than in the corresponding isopotential case. The intermediate potentiation patterns (SD_j = 0.5, turquoise filled circles) resulted in residual potentiations that were very similar to isopotential levels. We ran an additional set of simulations for the somato-centric distribution (SD_j = 0) in which AP generation was blocked (Fig. 6C, red open circles and dotted line). This revealed the significant contribution that the increased AP firing rate (attributable to the potentiation of proximal synapses) had to the weakening of the residual potentiations in the control simulation (Fig. 6C, solid red line). However, even in the absence of APs, the residual potentiation of the somato-centric potentiation patterns was still distinguishable from that of the peripheral patterns (compare red open circles with cyan filled circles). The lack of APs had no significant impact on the residual potentiations for the intermediate (SD_j = 0.5) and peripheral (SD_j = 1) configurations (data not shown).

We conclude that a g-HSP mechanism, based on activity at the soma, will weaken synaptic potentiations according to their spatial distribution throughout the dendrites. n_p potentiated synapses spatially distributed in a somato-centric pattern will be weakened more than an equal number of potentiated synapses in the corresponding isopotential neuron. Indeed, in an electrically distributed dendritic tree, the influence of proximal potentiated synapses, relative to the other nonpoteniated ones, is greater than in the corresponding isopotential case. For the same reasons, a peripheral pattern of potentiated synapses will be weakened less than in an isopotential neuron.

Under l-HSP, as we have seen (Figs. 4, 5A, middle, 5B, middle), the amount of weakening of potentiated synapses is mainly determined at the level of the dendritic branch, according to the degree of sparseness of the potentiated synapses on that branch. As a first approximation for this branch-specific sparseness, we chose the probability of potentiation, pLTP_k, of synapses on branch k. This measure is related to the ratio of the number of potentiated synapses to the total number of synapses, n_p,k/n_k, used to analytically predict the residual potentiation in the isopotential case (Fig. 1C). In dendritic trees, the pLTP_k we applied also reflects the spatial distribution of the potentiated synapses (e.g., with low pLTP_k, the chances that the potentiated synapses will be spatially segregated is high). In Figure 6D, we plotted the average residual potentiation, 〈rp〉, of the potentiated synapses from Figure 5A (middle) on each branch against the pLTP of that branch (red dots). The theoretical curve from Figure 1C, for an initial potentiation of p⁰ = 150%, is also plotted (red line). The match was remarkable (>70% of the variance of 〈rp〉 could be explained by the theoretical curve). We repeated this for an initial potentiation of p⁰ = 200% (green dots and line) and an initial depression of 50% (orange dots and line) and obtained similarly good matches with the theoretical curves. Thus, in each dendritic branch undergoing l-HSP, the degree of retention of n_p synaptic potentiations that are well spaced is equivalent to that of any n_p synaptic potentiations in an isopotential dendrite. However, as we have shown (Fig. 4), if the same number of synaptic potentiations are clustered, then the residual potentiation will be much smaller than in the isopotential case.

Finally, we sought a measure that could predict the residual potentiation of each individual synapse after l-HSP in the dendritic tree. In electrically distributed dendrites, not only n_p matters but also the distance of the n_p potentiated synapses one from another. Therefore, we defined an estimator, ss_i⁰ (supplemental Appendix II, Equation A12, available at www.jneurosci.org as supplemental material), for the residual potentiation of each potentiated synapse i, based on the spatial segregation of the initial potentiation of that synapse. This spatial segregation index, ss_i⁰, is an average of the ratio of the initial potentiation of synapse i and that of each one of its neighbors, j, weighted according to the distance between these synapses. Importantly, the definition of ss_i⁰ includes a free parameter, α (in micrometers), reflecting the size of the region that influences the fate of each potentiated synapse within the dendritic branch.

ss_i⁰ will equal 1 (predicting a complete loss of potentiation, rp = 100%) in either the case in which all synapses on the branch are equally potentiated (p_i⁰ = p⁰ ∀i) or the case in which there are only very few synapses on that branch. In contrast, ss_i⁰ will equal p_i⁰ (predicting full conservation of the initial potentiation, rp = p_i⁰) if synapse i is the only potentiated synapse on a crowded branch (the rest of the synapses have p_i⁰ = 100%).

In Figure 6E, the actual residual potentiation of each potentiated synapse (from Fig. 5A, middle) is plotted against the spatial segregation estimator, ss_i⁰ for α = 46 μm (red dots). The correlation was excellent (r² = 0.9). We also plotted the suppression of the nonpotentiated synapses against their ss_i⁰ and obtained an equally good correlation (blue dots). The value of α (supplemental Appendix II, Equation A12, available at www.jneurosci.org as supplemental material) was chosen so as to maximize the correlation coefficient, r². This “effective region of influence,” in the order of several tens of micrometers, seems to be typical for the interaction between synapses in dendritic branches (Engert and Bonhoeffer, 1997). Increasing the interval between the synapses (and thus decreasing their number) did not affect the optimal α because, as we mentioned, α reflects the size of the relevant dendritic region influencing synapse i and not the number of synapses occupying this region. In contrast, increasing the effective electrical space constant of the dendrites, λ (e.g., by reducing the axial resistance, R_i, or by “blocking” the voltage-sensitive ion channels in this branch) increased the optimal value for α (data not shown). In conclusion, under l-HSP, a simple distance-weighted average of the initial synaptic potentiations along the dendritic branch is sufficient for predicting the residual potentiation of each synapse.