Detailed Dendritic Excitatory/Inhibitory Balance through Heterosynaptic Spike-Timing-Dependent Plasticity

Calcium-based synaptic plasticity model with current-based heterosynaptic interaction explains h-STDP

We constructed a model of a dendritic spine, as shown in Figure 1A (see Materials and Methods, Spine model). In the model, the membrane potential of the spine u(t) is modulated by the current influx/outflux via the AMPA and NMDA receptors (Fig. 1A, x^A and g_N(u)x^N), spike backpropagation (x^BP), and heterosynaptic currents from nearby excitatory and inhibitory synapses (x^E and x^I, respectively; see Table 1 for the definitions of variables). The calcium concentration in the spine c(t) is controlled through the NMDA receptors and the VDCCs g_v(u) (Higley and Sabatini, 2012). Because both NMDA and VDCC are voltage-dependent (Lüscher and Malenka, 2012), the calcium level in the spine is indirectly controlled by presynaptic, postsynaptic, and heterosynaptic activities (Fig. 1B, top and middle). The voltage dependence of NMDA and VDCC [g_N(u) and g_v(u)] were assumed to be linear for simplicity. This linear assumption may overestimate the effect of a backpropagating action potential on the dendritic NMDA receptors in the overshooting phase, but the effect is expectedly insignificant because in the absence of Ca²⁺ spike, a backpropagated action potential tends to overshoot less at the dendrite compared with the soma due to dendritic attenuation (London and Häusser, 2005). In addition, although the synaptic input at an inhibitory synapse causes a positive Ca²⁺ influx by itself (Koch, 1998), activation of nearby inhibitory input hyperpolarizes the membrane potential at the local dendritic site, and may suppress the Ca²⁺ influx through VDCCs. Indeed, recent experimental results reveal that Ca²⁺ influx driven by backpropagating spikes or excitatory synaptic inputs is strongly reduced by an inhibitory input in a temporally and spatially precise manner (Hayama et al., 2013; Marlin and Carter, 2014; Müllner et al., 2015). On the basis of these observations, the Ca²⁺ level in our model is negatively regulated by the heterosynaptic inhibitory inputs through the hyperpolarization of the membrane potential. To model synaptic plasticity, we used a calcium-based model in which LTP/LTD are initiated if the Ca²⁺ level reaches above the LTP/LTD thresholds (Fig. 1B, middle, orange and cyan lines); this plasticity model reproduces the features of homosynaptic STDP very well (Shouval et al., 2002; Graupner and Brunel, 2012). We introduced the interim weight variable y(t) to capture the non-graded nature of synaptic weight change (Petersen et al., 1998). Thus, changes in Ca²⁺ level are first embodied in the interim weight y(t) (Fig. 1B, bottom), and are then reflected in the synaptic weight w(t) upon accumulation. The interim weight y(t) is expected to correspond with the concentration of active plasticity-related enzymes such as CaMKII or PP1 (Graupner and Brunel, 2007), and the synaptic weight w(t) reflects the somatic EPSP amplitude. Despite the modification, our model replicates the properties of homosynaptic STDP reproduced by Graupner and Brunel (2012) well (Fig. 1C,D).

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Figure 1.

Schematic figure of the model of h-STDP. A, A schematic figure of the model. Two variables in the spine, u(t) and c(t), represent the normalized membrane potential and Ca²⁺ concentration, respectively. Presynaptic action potentials modulate the membrane potential u(t) through AMPA (x^A) and NMDA (g_N(u)x^N) receptors. In addition, u(t) is modified by backpropagation (x^BP), and heterosynaptic current caused by excitatory (x^E) and inhibitory (x^I) inputs. The calcium level c(t) is modulated by influx/outflux through NMDA (g_N(u)x^N) and VDCC [g_V(u)]. Consequently, c(t) is indirectly controlled by u(t), because both NMDA and VDCC are voltage-dependent. B, An example of the dynamics of the membrane potential variable u(t) (top), Ca²⁺ concentration c(t) (middle), and the interim weight variable y(t) that controls the synaptic weight w(t) (bottom). The change in the Ca²⁺ level roughly follows the membrane potential dynamics, and the interim weight variable y(t) is positively (negatively) modulated when the Ca²⁺ level is above the LTP (LTD) thresholds represented by orange (cyan) dotted lines. Based on the variable y(t), the synaptic weight w(t) is updated on a slow timescale (Fig. 5C). C, Pairing frequency dependence of STDP. In corticostriatal synapse model without GABA, we changed the interval of pair-stimulation while keeping the total number of stimulations constant (see Materials and Methods). The blue 1 Hz line is the same as the red line in Figure 2A. The spike-timing windows for 20 and 40 Hz stimulations were shown only in the middle ranges because the spike-timing difference cannot be longer than the interval of stimulations, and the asymmetry around 0 ms is caused by the axonal delay. D, Dendritic position dependence of STDP. We mimicked dendritic position dependence by changing the amplitude of the backpropagation term γ_BP in the corticostriatal synapse model under the pre-post condition depicted by the red lines in Figure 2B. In the black line, the AMPA and NMDA coefficients were kept at the same values as in Figure 2B, whereas they were increased threefold in the gray line.

We first consider the effect of inhibitory input on synaptic plasticity at nearby excitatory spines. A recent experimental study in a medium spiny neuron (Paille et al., 2013) revealed that synaptic connections from cortical excitatory neurons typically show anti-Hebbian type STDP under a pairwise stimulation protocol, but if the GABA-A receptor is blocked, the STDP time window flips to a Hebbian type STDP (Fig. 2A, circles). Our model can explain this phenomenon in the following way. Let us first consider the case when the presynaptic excitatory input arrives before the postsynaptic spike (the “pre-post” regime). If the GABAergic input is blocked, presynaptic and postsynaptic spikes jointly cause a large membrane depolarization at the excitatory spine. After repetitive stimulation, the calcium concentration rises above the LTP threshold (Fig. 2B, top-right, red line), hence inducing LTP (Fig. 2B, bottom-right, red line). By contrast, if the GABAergic input arrives coincidentally with the presynaptic input, depolarization at the excitatory spine is attenuated by a negative current influx though the inhibitory synapse. As a result, the calcium concentration cannot reach the LTP threshold although it is still high enough to eventually cause LTD (Fig. 2B, right, black lines). Similarly, when the postsynaptic spike arrives at the spine before the presynaptic spike (the post-pre regime) in the absence of GABAergic input, the delayed presynaptic spike causes a slowing of the decay in the level of calcium concentration that may induce LTD (Fig. 2B, left, red lines). To the contrary, if the GABAergic input is provided simultaneously with the presynaptic input, the decay in the calcium concentration is sped up because of the hyperpolarization of the membrane potential at the excitatory spine by the inhibitory input. As a result, LTP is more likely to occur (Fig. 2B, left, black lines). Therefore, when a GABAergic input arrives in coincidence with a presynaptic excitatory input, the STDP time window changes its sign in both the pre-post and the post-pre stimulation regimes (Fig. 2A, lines).

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Figure 2.

The model reproduces spike-timing-dependent heterosynaptic effects. A, Spike timing window with/without a disynaptic GABAergic input. The lines represent simulation data, and the points are experimental data taken from Paille et al. (2013). Vertical dotted lines represent the spike-timing differences from which B was calculated. B, Dynamics of calcium concentration c(t) (top) and the interim weight variable y(t) (bottom) at the stimulated spine. Gray areas in the bottom figures represent the regions satisfying y(t) < y_th/K_rep, in which the change in the interim weight is not reflected to the synaptic weight, where K_rep represents the number of paired stimulations given in the simulation for A. C, Synaptic weight change with/without GABAergic inputs immediately before pre-post stimulation. Data points were taken from Hayama et al. (2013). The gray point is a result from muscimol application, not GABA uncaging. D, Dynamics of c(t) and y(t) at the stimulated spine. Black lines represent the dynamics under GABA uncaging, and red lines represent the controls. E, Synaptic weight change at a neighboring spine through excitatory heterosynaptic interaction. F, Dynamics of c(t) and y(t) at the neighboring spine.

A GABAergic effect on excitatory synaptic plasticity is also observed in CA1 pyramidal neurons (Hayama et al., 2013). In this case, the post-pre stimulation does not induce LTD unless GABA uncaging is conducted near the excitatory spine immediately before the postsynaptic spike arrives at the spine, whereas LTP is induced by the pre-post stimulation regardless of GABA uncaging (Fig. 2C, squares). Our model can also replicate these results. In the pre-post stimulation, the membrane potential of the spine shows strong depolarization due to positive feedback through the NMDA receptor, even if inhibitory current is delivered through GABA (Fig. 2D, top-right, red and black lines). Thus, LTP occurs after repetitive stimulation (Fig. 2D, bottom-right, red and black lines). By contrast, in the post-pre protocol, the effects of triggering LTP and LTD tend to cancel each other in the absence of GABAergic input, whereas LTD becomes dominant under the influence of GABAergic input (Fig. 2D, left, red and black lines, respectively).

In addition to the I-to-E effect, the E-to-E effect is also observed in the case of CA1 pyramidal neurons (Hayama et al., 2013). If GABA uncaging is performed immediately before postsynaptic firing, LTD is also observed in neighboring excitatory spines (Fig. 2E, right, point). This E-to-E heterosynaptic effect is not observed in the absence of GABAergic input (Fig. 2E, left, points). Correspondingly, in the model, excitatory current influx from a nearby synapse causes mild potentiation of calcium concentration in cooperation with inhibitory current influx, eventually inducing LTD (Fig. 2F, left, gray lines). Note that, in this E-to-E effect, interactions of signaling molecules or competition for resources at a later stage of synaptic plasticity may also play a dominant role (Hayama et al., 2013).

To check the parametric robustness of the model, we uniformly sampled the values of all the main parameters from fixed ranges, and studied the sensitivity of the model performance to each parameter, by calculating the performance distribution over the distributions of the other parameters (Fig. 3A,B). Even if the values of the parameters were perturbed by 20–100% from the original values (i.e., the values used in Fig. 2A), we obtained similar sizes of fitting errors to Figure 2A in ∼2.5% of the simulations (Fig. 3A). Moreover, we found that anti-correlation between the VDCC coefficient and the backpropagation coefficient was crucial for replicating the experimental result (Fig. 3B, α_V vs γ_BP and γ_BP vsα_V), as their product determined the effective amplitude of the calcium transient caused by a postsynaptic spike (Eq. 1 + Eq. 3). A linear relationship between the LTP constant and LTD constant is also important for model fitting (Fig. 3B, C_p vs C_d and C_d vs C_p). Notably, fitting the experimental data from the striatum requires a larger coefficient of the heterosynaptic inhibitory effect than the value required for fitting the data from CA1, and thus the striatum model depends on stronger inhibition than the CA1 model (Fig. 3C, top). This is consistent with the observation of strong inhibition in the striatum (Mallet et al., 2005). We also found that faithful reproducing of the CA1 experimental data crucially depended on a high NMDA/AMPA ratio, whereas the striatum model was rather robust against this ratio (Fig. 3C, bottom).

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Figure 3.

Parametric robustness of the h-STDP model. A, Fitting errors for the model of the striatum experiment at various values of model parameters. We conducted 10⁵ simulations by uniformly sampling all parameters within fixed ranges, and then calculated the median error (light gray line), 16 percentile error (dark gray line), and 2.5 percentile error (black line) for each parameter by marginalizing the other parameters. The x-axes show the range of perturbation used for all parameters, with the 100% values corresponding to the values used in Figure 2A (Table 2). Dotted lines represent the fitting error for the parameter set used in Figure 2A. See Materials and Methods for the definitions of the parameters. The fitting error was calculated as , where w_sim is the synaptic weight in the simulation, w_data is the weight in the experiment (Paille et al., 2013), and N_D is the number of data points (N_D = 64). B, Performance dependence on two-parameter relationships. For the same data depicted in A, we calculated the percentage of simulation trials that exceeded the fitting error criteria (error < 60.0), under a given parameter pair. Dark blue points are parameter sets at which 0% of the simulations exceeded the criteria, whereas red points are the parameter sets where 15% did. The x- and y-axes for each parameter are the same as in A. C, Comparisons of the parametric dependences of models fitted to the results from the striatum and CA1 experiments. The vertical dotted lines represent the values used in Figure 2. The NMDA/AMPA ratio in the bottom panel was calculated as γ_Nτ_N/γ_Aτ_A at various values of γ_N. Here, the error for the striatum model was calculated over 10⁴ simulations, and the NMDA coefficient γ_N was uniformly sampled from [0.0, 0.25). Parameters for the CA1 model were sampled from the same ranges. Three lines with different shades represent the fitting error at 6.7, 2.5, and 0.7 percentiles.

Phase transitions underlying h-STDP

In the previous section, we introduced a biophysical model to establish its relevance to the corresponding biological processes and obtain insight into the underlying mechanism. However, not all of the components of the model are necessary to reproduce the observed properties of h-STDP. We next provide a simple analytically tractable model to investigate the generality of the proposed mechanism.

To this end, we simplify the model to one in which the calcium level at a spine is directly modulated by the presynaptic, postsynaptic, and heterosynaptic activities, as given below: Here, C_i(t) represents the Ca²⁺ concentration at spine i, X_i, and X_post represent presynaptic and postsynaptic spikes respectively, d_I and d_E are heterosynaptic delays, and Ω_i^I and Ω_i^E are the sets of neighboring inhibitory and excitatory synapses (for details of the model, see Materials and Methods, Reduced model). Despite its simplicity, the model can qualitatively reproduce the heterosynaptic effects observed in striatal and CA1 neurons, although the quantitative accuracy is degraded (Fig. 4A and B, respectively). Importantly, the reduced model provides further analytical insights into the phenomena.

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Figure 4.

Phase transitions on the STDP time window in an analytical model of h-STDP. A, B, STDP windows at various values of heterosynaptic inhibitory effect C_I. A corresponds to the striatum experiment and B corresponds to the CA1 experiment. B, Top and bottom represent the stimulated and a neighboring spine, respectively. Note that values in B were calculated by ỹ = sgn(y) · [|y| − 15]₊ to reflect the effect of thresholding. C, Phase diagram of the STDP time window calculated for the inhibitory effect C_I and LTP threshold θ_p. The colors show the number of local minima/maxima, whereas the lines are typical STDP time windows at each phase. The parameters written on the right side (top) of the panel represent the critical values of θ_p (C_I). D, The phase diagram calculated for the heterosynaptic excitatory effect parameter C_E and the postsynaptic effect parameters C_post at a fixed inhibitory effect (C_I = 0.5). See Materials and Methods, Reduced model.

Let us consider how the inhibitory effect parameter C_I controls the I-to-E heterosynaptic effect observed in the CA1 experiment. If we characterize the shape of the STDP time windows by the total number of their local minima/maxima, the parameter space can be divided into several different phases (Fig. 4C). If the LTP threshold θ_p satisfies C_pre < θ_p < C_post, a Hebbian type STDP time window appears when the strength of heterosynaptic inhibitory effect C_I satisfies (C_post − θ_p)e^δ_I/τ_C < C_I < C_pree^δ_I/τ_C (Fig. 4C, top, orange region; see Materials and Methods for details of the analysis). Here, we defined δ_I as the spike-timing difference between the inhibitory spike and the presynaptic or postsynaptic spikes in the pre-post or the post-pre stimulation protocols, respectively. If C_I is larger than C_preexp(δ_I/τ_C), a strong inhibitory effect causes LTD, even in the pre-post regime (Fig. 4C, green region), whereas LTD in the post-pre regime is suppressed when C_I is smaller than (C_pre − θ_p) exp (δ_I/τ_C) (Fig. 4C, gray region). Thus, the heterosynaptic LTD observed in Figure 2C can be represented as the phase shift from the gray-colored region to the orange-colored region in Figure 4C, which is due to the change in the inhibitory effect C_I. This analysis further confirms the condition for inducing heterosynaptic LTD where the heterosynaptic spike-timing difference δ_I should be smaller than the timescale of Ca²⁺ dynamics τ_C (Hayama et al., 2013). This is because δ_l < τ_C is necessary for a significant heterosynaptic LTD, and C_I is typically smaller than C_post and θ_p. In addition, heterosynaptic suppression of the pre-post LTP (green region) is very unlikely to occur, as it is necessary for C_I to be larger than C_preexp(δ_I/τ_C). This condition is difficult to satisfy even if δ_I = 0, because the heterosynaptic effect on Ca²⁺ dynamics in the spine is expected to be smaller than the homosynaptic effect (i.e., C_I < C_pre). A similar analysis is possible for E-to-E interactions, although the phase diagram becomes complicated in this case (Fig. 4D; see Materials and Methods).

These analyses reveal that the heterosynaptic effects are always observable when the parameters of the calcium dynamics fall within a certain region of the parameter space, and underscore the robustness of h-STDP in our framework.

h-STDP induces the detailed dendritic E/I balance at dendritic hotspots

The results described so far suggest that the proposed model gives a good approximation of h-STDP. To investigate the possible functions of h-STDP, we next examined how this h-STDP rule shapes the synaptic organization on the dendrite of a simulated neuron. We first considered a model of a dendritic hotspot (Jia et al., 2010) that receives 10 excitatory inputs and one inhibitory input (Fig. 5A), because the heterosynaptic effect is typically confined within 10 μm of the synapse (Hayama et al., 2013). Excitatory inputs are organized into five pairs, with each pair of excitatory synapses receiving correlated inputs (Fig. 5B; see Materials and Methods, Dendritic hotspot model). In addition, the inhibitory input is correlated with one excitatory pair (Fig. 5A, A1 and A2). Here, we assumed that postsynaptic activity follows a Poisson process with a fixed rate, because the influence of a single hotspot on the soma is usually negligible. In addition, we neglected the effect of morphology and supposed that the heterosynaptic interaction occurs instantaneously within the hotspot. In this configuration, surprisingly, excitatory synapses correlated with the inhibitory input are potentiated, while other excitatory synapses experience minor depression (Fig. 5C, top). As a result, the dendritic membrane potential of the branch becomes less correlated with all the hidden signals, because the strong negative correlation with the blue signal is cancelled by the potentiated excitatory inputs, whereas weak positive correlations with other signals are diminished due to LTD at the corresponding excitatory synapses (Fig. 5C, bottom). Although input spikes to the hotspot is sparse and stochastic, the traces of excitatory and inhibitory currents shift toward the detailed balance after learning (Fig. 5D). A spike correlation between excitatory and inhibitory inputs is crucial for this potentiation of excitatory synapses, but in the model a small correlation is sufficient to produce a significant change in the synaptic weight (Fig. 5E). Moreover, this GABA-driven potentiation is only observable when inhibitory activity is precisely correlated with excitatory activities, and becomes larger when the inhibitory spike precedes excitatory spikes, rather than when it follow them (Fig. 5F). We also found that, when the heterosynaptic inhibitory effect γ_I is large enough to cause a strong hyperpolarization at nearby synapses, depression is observed at correlated excitatory synapses (Fig. 5G, blue area) instead of potentiation (Fig. 5G, red area). However, as can be seen in Figures 3 and 4, such a large inhibitory effect does not reproduce the STDP experiments, especially the data from the CA1 pyramidal neurons, and is thus unlikely to be observed in the actual brain. These results indicate that h-STDP induces a dendrite-specific temporally precise E/I balance by potentiating excitatory synapses that are correlated with inhibitory synapses.

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Figure 5.

Emergence of detailed dendritic excitatory/inhibitory balance by h-STDP. A, A schematic figure of a dendritic hotspot model. The shaft synapse represents an inhibitory input. Colors represent the spike correlations between synaptic inputs. B, Examples of correlated spike inputs. Each raster plot was calculated from 50 simulation trials. C, Changes in the synaptic weight, w (top), and the correlation between the dendritic membrane potential and hidden signals (bottom), under h-STDP. The blue lines represent the dynamics of synapses correlated with the inhibitory input. D, Traces of excitatory and inhibitory inputs before and after learning. Positive correlations between excitatory and inhibitory currents before learning (top, black and red lines) shift to large negative correlations after learning (bottom, lines). The excitatory and inhibitory input currents were approximately defined as u_E(t) = ∑_i=1¹⁰ w_i[γ_Ax_i^A(t) + γ_Ng_N(u_i)x_i^N(t)] and u_I(t) = −γ_Gx_G(t), respectively. In the graph, we scaled the inhibitory current twofold for illustration purposes. The green lines represent the dendritic membrane potential u_b(t). The learning was performed using the same parameter values as in the simulation depicted in C. E, Synaptic weight dependence on excitatory/inhibitory spike correlation. We changed the correlation by modulating the sensitivity of inhibitory activity to the hidden signal, while keeping the inhibitory firing rate fixed at r_I = 9.2 Hz. The spike correlation was calculated by taking 25 millisecond bins. F, Synaptic weight change at the excitatory synapses correlated with the inhibitory inputs (blue), and at other synapses (gray), at various inhibitory delays. Error bars in C, E, and F represent SDs over 50 simulation trials. G, The relative weight changes w_R calculated for various parameters. We defined w_R as 〈w_i^E〉_i∈corr − 〈w_i^E〉_{i∈un−corr}, where “corr” represents a set of excitatory synapses correlated with the inhibitory synapse, and “un-corr” stands for uncorrelated ones. The weights were calculated by taking an average over 10 simulations. H, The probability of a LTP/LTD occurrence after a presynaptic spike calculated from a simulation. The lines represent the mean LTP/LTD probabilities at excitatory synapses correlated with the inhibitory input (blue lines) and other synapses (gray lines), respectively. I, J, Results from single-spike simulations. The E/I coincidence prevents the LTD effect due to the pre-spike (I), without affecting the LTP effect due to pre-post coincidence (J). In I, the inhibitory spikes were provided at t = 0 for the black line and t = −100 ms for the gray line, with the excitatory presynaptic spike being given at t = 0 in both lines. Similarly, in J, the postsynaptic spikes were provided at t = −75 (light-gray), 0 (black), and +75 ms (dark-gray), while the presynaptic spike was given at t = 0 in all lines.

To reveal the underlying mechanism of this E/I balance generation, we used the simulation data to calculate the probability of the calcium level reaching above the LTD/LTP thresholds after a presynaptic spike. The probabilities of LTP occurrences show similar trajectories after a presynaptic spike, regardless of whether the presynaptic activity is correlated with inhibitory input or not (Fig. 5H, blue and gray dotted lines, respectively). However, the peak probability of LTD occurrence is significantly lower for spines that are correlated with inhibitory inputs (Fig. 5H, blue vs gray solid lines), although in both cases the probability goes up after the presynaptic spike. This asymmetry between LTP and LTD is consistent with the following interpretation: LTD is mainly caused when the presynaptic neuron fires at a low firing rate and the postsynaptic neuron remains silent, both in the experiments (Malenka and Bear, 2004) and in our model (Fig. 5I, gray line). However, if an inhibitory input arrives at a nearby dendrite in coincidence with excitatory activity, the calcium boost caused by the excitatory presynaptic input is attenuated by the heterosynaptic inhibitory effect (Fig. 5I, black line). As a result, LTD is shunted by correlated inhibitory inputs. On the other hand, LTP is mainly caused by coincident presynaptic and postsynaptic spikes, which induce a large increase in calcium that overwhelms the heterosynaptic inhibitory effect. Thus, LTP at correlated excitatory synapses is not compromised by inhibitory activity at a nearby site (Fig. 5J). Therefore, correlated spines tend to be potentiated overall.

To evaluate the generality of the observed dendritic E/I balance, we extended the model to a two-layered single cell (Poirazi et al., 2003) by modeling each branch with one dendritic hotspot (Fig. 6A; see Materials and Methods, Two-layered neuron model), and investigated the dendritic organization of synaptic weight changes by h-STDP. In the simulation, we introduced a 10 ms delay between the excitatory and inhibitory stimulation (Froemke, 2015). Even in this case, when the dendritic branches of a postsynaptic neuron receive inputs from various neurons with different selectivity, each dendritic hotspot shapes its excitatory synaptic organization according to the selectivity of its inhibitory input (Fig. 6B,D; the frame colors in B represent the inhibitory selectivities). As a result, the excitatory synapses on the dendritic tree become clustered, as observed in previous experiments (Kleindienst et al., 2011; Takahashi et al., 2012). Note that, in our model, this clustering of excitatory synapses is caused by common inhibitory inputs, instead of direct interactions between excitatory spines.

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Figure 6.

Detailed dendritic excitatory/inhibitory balance in a two-layered single cell model. A, A schematic illustration of the single cell model. The actual model has 100 dendritic branches each receiving 10 excitatory inputs and 1 inhibitory input. As in Figure 5A, the inhibitory inputs are represented by shaft synapses. B, Examples of synaptic weight change at each branch. The color of the frames represents the selectivity of the inhibitory input to the branch. Each row represents a different simulation trial. C, An example of synaptic weight change at each branch under the standard STDP rule. As in B, the frame colors represent the local inhibitory selectivity. D, The mean synaptic weight dynamics of synapses correlated with the local inhibitory inputs and other synapses under h-STDP. E, Development of the branch dissimilarity under the two learning rules. Dissimilarity was given as . F, Raster plots of output spikes before (top) and after (middle) learning, and their firing rate dynamics (bottom), taken from 100 trials each for 10 simulated neurons. The colors of the spikes in the raster plots represent results from different simulation trials. The black vertical lines represent the points of change in excitatory inputs, and the horizontal colored bars at the top of the bottom panel correspond to the colors of the presynaptic neurons active in each period. G, The ratio of change detecting spikes before and after learning. The ratio was defined as the fraction of spikes occurring within 50 ms of a change in stimuli to the total number of spikes. On the x-axis, in addition to the mean excitatory input firing rates, the mean inhibitory input firing rates were also modulated from 50 to 210 Hz correspondingly, to maintain the E/I balance of the input. H, The ratio of change detecting spikes with/without inhibitory shuffling. The gray line is the same with the black line in G. In the black line, selectivities of inhibitory synapses were randomly shuffled to break the E/I balance at each dendrite. We introduced a 10 ms delay between the excitatory and inhibitory stimulus, both during learning (B–E) and in the change detecting task (F–H). The averages in D, E, G, and H were taken over 10 simulation trials.

To further clarify the importance of heterosynaptic interaction, we next compared the results of h-STDP with that of learning under the standard STDP (Song et al., 2000). In the model, we introduced a branch-specific homeostatic plasticity to induce competition among nearby synapses (see Materials and Methods for the details of the standard STDP model). In the standard STDP model, all dendritic branches developed similar synaptic distributions regardless of the differences in local inhibitory selectivity (Fig. 6C; the beige signal is learned in this example), because one of the hidden signals is captured by chance through self-organization (Song et al., 2000). As a result, under the standard STDP rule the synaptic organizations of branches remain akin to each other (Fig. 6E, gray line). By contrast, under the h-STDP rule, each dendritic branch acquires a synaptic structure according to its local inhibitory input, and individual dendritic branches become dissimilar from one another (Fig. 6E, black line).

We further investigated the possible function of this synaptic organization in information processing. To this end, we consecutively presented the five stimuli to the two-layered neuron model (Fig. 6F). Before learning, the neuron showed an almost constant response to the stimulation, with a small dip at the change points between each of the five stimulations (Fig. 6F, top). By contrast, after learning, the neuron showed transient bursting activity immediately after the onset of the new 500 ms stimulus window, and then rapidly returned to an almost silent state (Fig. 6F, middle). Hence, by h-STDP, a neuron can acquire sensitivity toward abrupt changes in stimuli (Fig. 6F, bottom, G). This sensitivity vanished if the selectivities of inhibitory synapses were randomly shuffled, suggesting the importance of the balance at each dendrite (Fig. 6H). This result indicates that although the detailed E/I balance at each dendrite has a small overall effect to the somatic membrane dynamics; the collective effect from all dendritic branches has a significant impact on the postsynaptic activity.

The effect of dendritic spikes

Previous studies on dendritic computation reveal the potential importance of dendritic spikes in synaptic plasticity (Smith et al., 2013; Kastellakis et al., 2016). Although we mainly consider a regime within a low firing rate (∼5 Hz), dendritic spikes may occur due to strong input spike correlation. We therefore extended the model discussed in the previous section by including dendritic spikes.

Here, we focus on Na⁺ spikes that are typically localized within a dendritic branch, not global Ca²⁺ spikes (London and Häusser, 2005), as we are interested in branch-specific synaptic organization. We modeled dendritic spikes as an excitatory heterosynaptic interaction with thresholding, and then added the interaction to the dendritic hotspot model (see Materials and Methods, Dendritic hotspot model). When the amplitude of a dendritic spike is comparable to the amplitude of backpropagating spike, the synapses set their weights differently, depending on the threshold of a dendritic spike. When the spike threshold is very low, the relative weight difference between the correlated and uncorrelated excitatory synapses converges to zero, because all excitatory synapses are potentiated in this regime. In contrast, under conditions of a very high threshold, dendritic spikes rarely occur, so the correlated synapses are only moderately potentiated, as in the control (Fig. 7A; the dotted line represents the control). Notably, when four to five spikes from synapses with the unit weight (w = w_o) are sufficient to generate dendritic spikes, the relative weight differences becomes larger than the control, as the dendritic spikes selectively strengthen such synapses that are moderately potentiated by h-STDP (Fig. 7A). This effect is especially significant when inhibitory spikes are delayed by 5–10 ms on average (Fig. 7B). These results suggest that local dendritic spikes stabilize the synaptic weight structure generated through heterosynaptic STDP.

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Figure 7.

Dendritic spike stabilizes the detailed dendritic balance. A, Relative synaptic weight w_R obtained at various dendritic spike thresholds for an inhibitory delay of 10 ms. The relative synaptic weight was defined as the difference between the weights of the synapses correlated with the inhibitory input and the other excitatory synapses, as in Figure 5G. The x-axis is normalized as θ̂_ds = θ_ds/w_o, and the dotted line shows the value without dendritic spikes (control). B, Relative synaptic weight w_R for various inhibitory delays at θ̂_ds = 4.0. The gray line is the same as the difference between the two lines in Figure 5F.

h-STDP explains the critical period plasticity of binocular matching

The results so far indicate that h-STDP induces GABA-driven reorganization of synaptic weights, which in turn enriches dendritic computation such as in the enhanced sensitivity to detect changes in input activity. To investigate its relationship with developmental plasticity, we next consider a model of critical period plasticity in binocular matching (B. S. Wang et al., 2010, 2013). In mice, 1 week after the eye opening, binocular neurons in V1 typically exhibit different orientation selectivity for inputs from the two eyes. Nevertheless, after another 2 weeks, the selective orientations for each eye become closer, and eventually they almost coincide with each other (B. S. Wang et al., 2010). Moreover, this binocular matching is disrupted by accelerating inhibitory maturation (B. S. Wang et al., 2013). Thus, the activity of inhibitory neurons plays a decisive role in shaping binocular matching, in addition to Hebbian plasticity at excitatory synapses.

We modeled this process with the two-layered single cell model introduced in Figure 6 (Fig. 8A, right; see Materials and Methods, The model of binocular matching). The input spike trains were modeled as rate-modulated Poisson processes driven by a circular variable θ, which corresponds to the direction of moving visual stimuli. We assumed the following: (1) inputs from ipsilateral and contralateral eyes already have some weak orientation selectivity at the eye-opening (B. S. Wang et al., 2010; Espinosa and Stryker, 2012), (2) inhibitory cells are driven by both ipsilateral and contralateral eyes (Yazaki-Sugiyama et al., 2009; Kuhlman et al., 2011), and (3) the average orientation selectivity of inhibitory inputs fall between the orientation selectivity for ipsilateral and contralateral excitatory inputs (Fig. 8A, left). This last assumption has not yet been supported by experimental evidence, but if inhibition is provided by neighboring interneurons, these inhibitory neurons are likely to be driven by similar sets of feedforward excitatory inputs to those driving the output neuron. For mathematical convenience, we consider direction selectivity instead of orientation selectivity, but the same argument holds for the latter.

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Figure 8.

h-STDP can trigger binocular matching. A, Left, Direction selectivity of input neurons. In the model, as depicted by the black vertical lines, the majority of excitatory input neurons from the contralateral (ipsilateral) eye are selective for directions around θ = −π/4 (π/4, whereas inputs from the inhibitory neurons are weakly selective for θ = 0. Right, A schematic figure of the model configuration. Each dendritic branch receives inputs from both ipsilateral and contralateral-driven excitatory neurons, and also from inhibitory neurons. B, Top, Difference between the mean excitatory direction selectivity and inhibitory direction selectivity in each branch. Middle, Difference between the mean ipsilateral-driven excitatory direction selectivity and the mean contralateral-driven excitatory direction selectivity over all synapses on the neuron. Bottom, DSI calculated for contralateral inputs (purple), ipsilateral inputs (light green; hidden under the purple line), and binocular inputs (black). See Materials and Methods, The model of binocular matching for details of the evaluation methods. Red vertical lines represent the timings for the introduction of inhibitory inputs. Throughout Figure 8, the error bars are SD over 10 simulation trials. C, Firing responses of the neuron for monocular inputs, immediately after the initiation of inhibitory inputs (left; t = 30 min) and after the learning (right; t = 60 min). D, Examples of the direction selectivity of three representative branches before (gray lines; t = 0 min) and after (purple/light green lines; t = 60 min) the learning. Black lines represent the selectivity of the inhibitory input to the branch. E, Behavior in the monocular deprivation model. In the shadowed areas, contralateral-driven inputs were replaced with rate-fixed Poisson inputs to mimic monocular deprivation. The ordinates are the same with B. F, Synaptic weights development at different mean inhibitory selectivity. Ordinates are the same as in B, and the values were calculated at t = 60 min. The purple and green vertical dotted lines are the mean selectivity of contralateral and ipsilateral excitatory inputs, respectively.

In the simulation, we first ran the process without inhibition, and then introduced GABAergic inputs after a while (Fig. 8B,E, red vertical lines represent the starting points of inhibitory inputs), because maturation of the inhibitory neurons typically occurs in a later stage of the development (Hensch, 2005). Upon the introduction of inhibition, the mean preferred direction selectivity of excitatory synapses in each branch converges to that of the local inhibition, because of heterosynaptic plasticity (Fig. 8B, top; see Materials and Methods for details of evaluation methods), although the synaptic weight development was biased toward the overall direction selectivity of the postsynaptic neuron (Fig. 8D; the bias is toward the zero-degree direction). This dendritic E/I balancing reduces the difference between the direction selectivity of ipsilateral and contralateral inputs on average, because both become closer to the selectivity of the inhibitory input (Fig. 8B, middle). As a result, the binocular direction selectivity is strengthened (Fig. 8B, bottom), and the responses for monocular inputs approximately coincide with each other (Fig. 8C, right). Deprivation of the contralateral inputs immediately after the introduction of inhibition blocks binocular matching (Fig. 8E), in accordance with the experimental data (B. S. Wang et al., 2010).

Precocious GABA maturation has been reported to disrupt binocular matching (B. S. Wang et al., 2013). Our model suggests that the disruption is possibly related to the violation of the third assumption in the model. When the mean inhibitory direction selectivity is substantially different from the ipsilateral and the contralateral direction selectivity (Fig. 8F, at the parameter regions outside of the area surrounded by purple and green lines), h-STDP does not work effectively (Fig. 8F, top), and the difference between ipsilateral and contralateral inputs is not reduced (Fig. 8F, middle). As a result, the binocular direction selectivity is not improved by learning (Fig. 8F, bottom). These results indicate that the rate of maturation of GABA inputs and their effect on h-STDP are an important part of the underlying mechanisms of binocular matching in critical period plasticity.