All simulations were designed and run using NEURON 5.8 (Hines and Carnevale, 1997). Four neuron models, listed below, were used in this work. Each model contained calcium dynamics, inhibitory GABAergic synapses and excitatory AMPA/NMDA synapses undergoing [Ca²⁺]_i-dependent plasticity as in Shouval et al. (2002). A typical simulation time of 60 s was sufficient so that no further plasticity changes were observed. The integration time step in all simulations was Δt = 0.025 ms.

Modeled excitatory synapses had a combination of AMPAR- and NMDAR-dependent conductances, as in Larkum et al. (2009). We modeled the kinetics of the NMDA current by (1)gNMDA=gmax(e−t/τ1−e−t/τ2)/(1+0.25e−0.08V)​, where g_NMDA is the NMDA conductance and g_max is the peak synaptic conductance, τ₁ = 90 ms and τ₂ = 5 ms. The NMDA channels were assumed to be the sole source of Ca²⁺ current, as in Shouval et al. (2002).

AMPA current was modeled with an instantaneous rise time and an exponential decay time constant of 2 ms.

Synaptic plasticity followed a [Ca²⁺]_i-based learning rule, adapted from Shouval et al. (2002): (2)w˙j=η([Ca2+]i) (Ω([Ca2+]i)−wj), where w_j represents the strength of synapse j, the calcium level is denoted by [Ca²⁺]_i and η is the learning rate. Synaptic weights were updated at each time step during the simulations. The change in the AMPA and NMDA conductance values was proportional to the change in w_j. According to the learning function, Ω, synaptic plasticity depends on the level of [Ca²⁺]_i relative to two thresholds: the threshold for depression (LTD), θ_d, and the threshold for potentiation (LTP), θ_p. When the calcium level is below θ_d, no synaptic modification occurs; if θ_d < [Ca]_j < θ_p, the strength of the excitatory synapse, w_j, is reduced; and for [Ca]_j > θ_p, the synaptic strength is increased (Figure 1A). The learning rate, η, is assumed to depend on [Ca²⁺]_i, increasing monotonically with calcium levels (Figure 1B). The Ω function used throughout this paper was as in Shouval et al. (2002): (3)Ω=0.25+sig([Ca2+]i−α2, β2)−0.25sig([Ca2+]i−α1, β1), where (4)sig(x, β) = exp(βx)/(1+exp(βx)), with α₁ = 0.35, α₂ = 0.55 and β₁ = 80, β₂ = 80.

The calcium-dependent learning rate, η, was inversely related to the learning time constant, η = 1/τ. The functional form of τ was as in Shouval et al. (2002): (5)τ = P1/(P2+[Ca2+]iP3)+P4, with P₁ = 0.1 s, P₂ = P₁/10⁻⁴, P₃ = 3, and P₄ = 1 s.

In most of the simulations, excitatory synapses received synchronous inputs, except in Figure 6, where input activation times were drawn from a random Poisson process with a mean rate of 7 or 10 Hz.

Calcium dynamics included a calcium pump: (6)ICaP=I¯CaP/(1+(KMCaP/([Ca2+]i −[Ca2+]basal))), where I_CaP is the Ca²⁺ current flow through the pump (in mA/cm²), Ī_CaP is the maximal current flow through the pump, K_MCaP is a constant (0.05 mM), and [Ca²⁺]_basal is the basal level of intracellular calcium (0.25 μM).

Simulation of longitudinal diffusion between adjacent compartments was via a published NEURON mechanism for calcium diffusion (Carnevale and Hines, 2006). Longitudinal diffusion was modeled by: (7)d[Ca2+]idt=2×fCa (ICaP) / dsF, where f_Ca is the constant fraction of free calcium, d_s is the saturation diffusion coefficient of calcium, and F is Faraday's constant.

We first explored the interaction between inhibition and synaptic plasticity in the ball-and-stick model (Figure 1C; see “Materials and Methods”). Inhibition was activated on the dendritic compartment, and 21 NMDA- and AMPA-receptor-mediated excitatory synapses excitatory synapses received synchronous input at 10 Hz each, and underwent plasticity according to the learning rule (Equations 2–5). In the absence of inhibition (black line in Figure 1D), the calcium concentration along the dendritic cable varied with location, i.e., [Ca²⁺]_i gradually decreased toward the somatic compartment, due to the large sink imposed by the soma. These differences in [Ca²⁺]_i along the cable were accentuated when inhibitory input was activated, since local inhibition reduced depolarization and thus decreased calcium levels (Figure 1D, orange line). In accordance with calcium levels, synapses close to the soma and in the vicinity of inhibition were depressed. At the locus of inhibition (X = 0.6) the weight of the excitatory synapse was w_exc = 0 after 60 s of simulation (Figure 1E, cyan; colors corresponding to Figure 1C), whereas further away from inhibition (X = 1.6), toward the tip of the dendrite, the excitatory synapses were potentiated (Figure 1E, red). Accordingly, [Ca²⁺]_i near the location of inhibitory input was decreased to its basal level (Figure 1E, light cyan trace), whereas near the end of the cable, [Ca²⁺]_i levels rose significantly (Figure 1E, light red trace).

Synaptic inhibition, as well as dendritic geometry, affects the spatial dendritic profile of [Ca²⁺] which, in turn, determined the distribution of synaptic weights in the modeled dendrite. The [Ca²⁺]_i distribution on the dendritic cable depended on the strength of the inhibitory input. Different levels of GABAergic conductance were simulated using different clusters sizes of inhibitory connections (see “Materials and Methods”). Different spatial profiles of [Ca²⁺]_i were generated (Figure 2A), resulting in various distributions of synaptic weights, as depicted following 0, 6, 12, or 60 s of simulation (Figures 2B–D). When inhibition was activated at X = 0.6 (as in Figure 1), with a relatively small cluster of GABAergic conductances (total g_GABA = 5 nS, Figure 2B) synapses in the vicinity of inhibition switched from LTP (black line—without inhibition) to LTD (orange lines), whereas synapses located far from inhibition, toward the end of the cable, remained as potentiated as in the absence of inhibition (black line) after 60 s of simulations (continuous orange line). The spatial profile of w_exc reflected the shape of the Ca²⁺-dependent synaptic learning rule (Ω function in Figure 1A). With 5 nS (Figure 2B), only the two right-hand states of the Ω function—the LTD and the LTP states—were mapped onto the dendritic spatial profile of w_exc (Figure 2B, inset). With a larger inhibitory conductance change (g_GABA = 10 nS, Figure 2C), synapses on the left part of the cable did not undergo plasticity and remained at w_exc = 1 (protected), since in this case the intracellular calcium level in the dendritic region spanning the soma (X = 0) and about X = 0.3 was too low to induce plasticity (Figure 2A, green line), and the synaptic weights in this region remained unchanged throughout the simulation (Figure 2C). Near the location of the inhibitory input, excitatory synapses were depressed, whereas further away from the location of inhibition, the excitatory synapses were potentiated. In this case, the entire Ω function was mapped onto the dendritic spatial profile of w_exc (Figure 2C, inset). Finally, with a very high g_GABA value of 15 nS, most of the excitatory synapses remained protected, while away to the right of the inhibition, excitatory synapses were depressed (Figure 2D). In this case only the two left-most states of the learning rule curve—protected and LTD—were mapped onto the dendritic cable (Figure 2D, inset).

In the above simulations we used the simpler case of synchronous excitatory inputs. However, the results obtained for this case were also reproduced when the excitatory synapses were activated randomly (not shown). Moreover, to test whether the large gradient of calcium concentration along the cable was the result of the steep transition between LTD and LTP (Shouval et al., 2002), we changed Ω to achieve a more gradual transition between the three states (see “Materials and Methods”). The smoother learning function did not alter the qualitative results obtained above (not shown).

The spatial extent and particular profile of synaptic strengths strongly depended on the location of inhibition and on the cable properties of the dendrites. This is demonstrated in a model of a reconstructed L5 pyramidal cell, focusing on three different dendritic branches (Figures 3 and 4). In each case, we replaced the respective dendritic branch with a model composed of a cylinder (with a diameter and passive properties of that branch) and an isopotential compartment representing the measured load (the boundary conditions) imposed on the modeled branch by the rest of the tree (see “Materials and Methods”). Excitatory synapses were evenly distributed over the dendritic branch and inhibitory input with a conductance of g_inh = 1 nS was activated in various locations on that branch. We activated synaptic input on single branches, while the rest of the cell remained silent, in order to explore the impact of specific morphological and physiological conditions on synaptic plasticity. Although the conditions of such restricted input are usually not observed experimentally, we found them to be important tools for the understanding of the complex interaction between multiple excitatory and inhibitory inputs as depicted in Figure 6 below. When the inhibitory source was placed in the middle of a small distal apical branch, it induced two distinct plasticity states over different parts of this cable (Figure 3A; red—LTP, cyan—LTD), despite its short electrical length (L = 0.14λ). Such dissimilarity between the weights of identical adjacent synapses is the result of the large difference between the boundary conditions at the two ends of the branch, i.e., a large sink (small ρ, see “Materials and Methods”) at the proximal end (ρ = 0.01 in the modeled tree) and a sealed end at the distal portion of the dendritic branch. This asymmetry gives rise to a large [Ca²⁺]_i gradient along the branch in the absence of inhibition (Figure 3B, black). Inhibition impinging on this branch shifted the spatial distribution of [Ca²⁺]_i in a manner that depended on the strength and exact location of the synaptic inhibition. With inhibition at the proximal end (X = 0, Figure 3C, orange), most of the synapses underwent LTP, whereas synapses near the branch point experienced LTD. Inhibition at the center of the branch (X = 0.07, green) resulted in approximately half of the branch's synapses being depressed, while the synapses at the distal part were potentiated. With inhibition at the distal end (X = 0.14, purple), all of the branch's synapses underwent LTD. Thus, tuning the magnitude and/or the location of GABAergic dendritic synapses could serve as a mechanism for fine tuning the plasticity profile over the dendritic tree. Surprisingly, excitatory plasticity is tuned by local dendritic inhibition in an extremely precise manner, with micrometer resolution.

The degree of inhibitory tuning of the plasticity profile depended on the morphological properties of the dendrite. For example, in the relatively long main apical trunk (Figure 3D), the current sink at both ends (ρ₁ = 0.27 at the distal end and ρ₂ = 0.44 at the proximal end) lowered [Ca²⁺]_i near the branch ends relatively to its center (Figure 3E, black). Consequently, without inhibition, synapses near the branch ends were weakened (LTD), whereas synapses in the middle were strengthened (Figure 3F, black line). The addition of inhibition resulted in LTD at the center (Figure 3D, cyan), whereas synapses at both ends became protected (black). Thus, inhibition lowered [Ca²⁺]_i over the whole branch (Figure 3E) and switched the plasticity states of all of the excitatory synapses (Figure 3F). Although this branch was longer than the branch depicted at Figure 3A, the specific location of inhibition did not affect the spatial distribution of synaptic weights (orange, green, and purple lines). The similar boundary conditions at both ends governed the location of the transition point between plasticity states, and inhibition could only determine whether one or two plasticity states would emerge in this branch, and which two states were to exist throughout the branch (LTP/LTD or LTD/protected).

We concluded that the structural properties of the branch, predominantly its boundary conditions (taking into account the excitatory activity), set the spatial profile of calcium concentration along the branch, thus defining the range of possible synaptic weight distribution profiles that could be induced by the activation of inhibition.

Inhibition was found to be more effective in carving the synaptic weight distribution at a high spatial resolution when the slope of [Ca²⁺]_i along the dendrite was steep, as in the case of terminal branches (with a large sink at the proximal end), and this expression of the three plasticity states within a single branch is expected to occur in long branches with a large load on one end. This is the case depicted in the basal terminal dendrite (Figure 4A; black—protected, cyan—LTD, and red—LTP) where L = 0.4λ and ρ = 0.017. In this case three plasticity states existed simultaneously along the branch. In a simplified model of this branch (Figure 4B), the large dendritic load, emulated by an isopotential compartment with a corresponding ρ value, generated a sufficiently large [Ca²⁺]_i gradient (Figure 4B) for inhibition to change the location of both transition points between the three plasticity states along this branch (Figure 4C).

We next examined synaptic plasticity on this single basal branch during the activation of excitatory synapses (2321 non-plastic synapses) over the whole modeled neuron. We uniformly distributed 45 excitatory synapses on this branch and activated them simultaneously at 10 Hz. The rate of excitatory synaptic activity over the entire cell was varied. When excitation was activated on the depicted basal branch alone, the dendritic tree served as a current sink at the proximal end of the basal branch (Figures 4A–C). In contrast, strong activity (10 Hz for all synapses) served as a current source for this basal branch, leading to the strengthening of excitatory synapses at the proximal end (Figure 4D, red). In this particular simulation, in order to avoid saturation of LTP due to very high [Ca²⁺]_i along the branch we decreased the peak excitatory synaptic conductance on this branch by a factor of 2. We emulated this high activity state in a simplified “ball-and-stick” model by activating 50 excitatory synapses at the isopotential compartment, each at a rate of 10 Hz input (Figures 4E,F). The synaptic conductance, g_max, of the excitatory synapses in the dendritic compartment was decreased by a factor of 2, as in Figure 4D. The voltage source led to an increase in [Ca²⁺]_i near the proximal end of this branch (Figure 4E), resulting in the potentiation of the proximal synapses (Figure 4F), rather than them being protected in the low activity case (Figure 4C). More distal synapses experienced LTD under these conditions (Figure 4F). Thus, both the degree of excitatory synaptic activity, and inhibition strength and location, determined the spatial distribution of the synaptic weights. Therefore, various weight distributions become possible under different conditions of excitatory and inhibitory activity.

Next inhibitory input was placed on a single basal dendrite and excitatory synapses were activated on that same branch and on its parent and sister branches (Figure 5A, inset). We found that the “fate” of plasticity of the excitatory synapses depended on their branch location: for g_inh = 2 nS, the strength of the excitatory synapses remained unchanged in the immediate vicinity and distal end of the inhibited branch (black), whereas more proximally on this branch, LTD was induced (cyan). The synapses on the parent branch were also depressed but excitatory synapses on the sister branch were potentiated (red). Using an equivalent Y-shaped neuron model (insets in Figures 5C,D and see “Materials and Methods”) we activated inhibition on one of the distal locations (I). As g_inh increased from 2 nS (Figure 5B, green) to 20 nS (Figure 5B, orange), a larger area of the inhibited branch was rendered protected (w_exc = 1), more synapses switched from LTP to LTD in the inhibited branch and its parent branch, and LTP amplitude decreased in the sister branch.

Figures 5C–E plots the weights of excitatory synapses situated at different loci as a function of increasing inhibitory conductance at a constant location. At the locus of inhibition (Figure 5C), the excitatory synapse reached the non-plastic state very rapidly—low g_inh was sufficient to reduce calcium levels below the LTP threshold. Above g_inh = 10 nS, the calcium level was too low to induce plasticity. Parent synapses (Figure 5D) did not switch to the non-plastic stage at any value of g_inh, but rather switched from LTP to LTD. Synapses on the sister dendrite (Figure 5E) were always potentiated, independently of the magnitude of the inhibitory input. Insets depict the portion of the Ω function that is implemented in the depicted locations of the excitatory synapses, as a function of g_inh. In the inhibited branch all three states existed, at the parent only two states were possible (LTP/LTD), and at the sister branch only LTP was found.

Individual inhibitory neurons typically innervate pyramidal cells via multiple synaptic contacts; each class of inhibitory neurons targets different subdomain of the post-synaptic dendritic tree (Markram et al., 2004; Gidon and Segev, 2012). We therefore expanded our analysis to explore the implications of such inhibitory connectivity pattern on excitatory plasticity (Figure 6). Four domain-specific patterns of inhibitory innervation were examined: (1) Axo-axonic (Chandelier) inhibition (Somogyi, 1977; Somogyi et al., 1982; DeFelipe et al., 1985); (2) Peri-somatic (basket cell) inhibition forming multiple inhibitory contacts onto the soma and proximal basal and apical dendrites of the target pyramidal cells (Somogyi et al., 1983; DeFelipe et al., 1999); (3) Distal dendritic inhibition, such as formed by Martinotti cells onto the oblique and apical branches (Wang et al., 2004; Silberberg and Markram, 2007); and (4) Sparse inhibitory connections onto different dendritic regions, such as those formed by double bouquet (DB) inhibitory cells (Somogyi and Cowey, 1981; DeFelipe et al., 1989, 1999).

Axo-axonic inhibition, modeled by 25 inhibitory synapse of g_GABA = 1 nS onto the initial segment of the axon (Chandelier-like inhibition, Figure 6A), modified synaptic weights in the peri-somatic region. LTD was induced around the soma for higher excitatory activity (10 Hz; middle), while lower activity (7 Hz; right) resulted in LTD in the main apical trunk and a large peri-somatic non-plastic area. The neuronal activity level was critical in determining the span of inhibition's effect on plasticity, since it set the sink level for each dendritic branch. High activity served as a voltage source for each branch, resulting in a localized inhibitory effect on synaptic weights and in the absence of protected synapses. In contrast, a low rate of background synaptic activity generated a sink at each bifurcation point, leading to a decrease in [Ca²⁺]_i and lowered synaptic weights at a peri-somatic zone. This result is consistent with the findings of Gidon and Segev (2012), namely that given multiple synaptic contacts, inhibition operates globally, spreading centripetally up to hundreds of micrometers from the location of the inhibitory synapses.

For multiple inhibitory contacts onto the soma and basal branches, simulated by 25 inhibitory synapses with g_GABA = 1 nS (basket cell inhibition, Figure 6B), the range of inhibitory influence and the degree of plasticity modification was very similar to that of the axo-axonic inhibition for both high and low activity levels in the cell. This result is in accordance with previous findings, showing that due to a centripetal effect, peri-somatic dendritic inhibition generates an almost identical profile of shunt level (namely, the shunting effect of inhibition on the dendrites) over the cell as axo-axonic inhibition (Gidon and Segev, 2012).

Dendritic inhibition targeting both apical and basal branches, simulated by 25 synapses of g_GABA = 1 nS each (Martinotti inhibition, Figure 6C) enabled modulation of plasticity in both basal and apical micro domains (middle). As the background excitatory rate decreased to 7 Hz (right), the centripetal effect enabled inhibition to affect a substantially wider area, inducing a modification of excitatory synaptic weights throughout the apical tree.

Sparse inhibitory connections onto two dendritic domains, as formed by a single DB inhibitory cell, were simulated by 10 apical and basal contacts, each of g_GABA = 1 nS (Figure 6D). This type of innervation modified plasticity in a highly localized manner restricted to the source branch of inhibition, at both high and low background activity rate. At a lower rate of activity (7 Hz; right), the extent of weight modification widened, yet it remained much more localized than for the other patterns of inhibitory innervations (Figures 6A–C, right). Thus, sparse inhibitory contacts may override the centripetal effect at low rates of excitatory input. We suggest that fine tuning of synaptic plasticity may be achieved by either a high excitatory activity level or by sparse inhibitory innervations over the dendritic tree.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.