Introduction

Learning is thought to be implemented by several processes in the nervous system, one of which is synaptic plasticity - the modification of synapses based on incoming stimuli to the neuron (Citri and Malenka, 2007; Kandel et al., 2014; Takeuchi et al., 2014; Abraham et al., 2019). Most often, incoming stimuli cause calcium influx in the synapse, an important “second messenger” ion that triggers synaptic plasticity. Importantly, not all levels of calcium can elicit synaptic plasticity - threshold levels of calcium have been reported above which plasticity is triggered (Yang et al., 1999; Nishiyama et al., 2000; Cormier et al., 2001; Shindou et al., 2011). These calcium thresholds for inducing plasticity are themselves modifiable (plastic) through a process called metaplasticity (Abraham, 2008). Metaplasticity regulates the conditions under which plasticity occurs, i.e. under which synapses are modified.

In many neurons, such as hippocampal CA1 neurons, Purkinje neurons, cerebellar Golgi cells, and projection neurons from the core of the nucleus accumbens, the calcium signal for synaptic strengthening (long-term potentiation, LTP) and for synaptic weakening (long-term depression, LTD) comes from different sources (Nishiyama et al., 2000; Kohda et al., 1995; Binda et al., 2016; Locatelli et al., 2021; Ji and Martin, 2012; Cho et al., 2001). In the direct-pathway striatal projection neurons (dSPNs), a model of which is used in this study, calcium from NMDA receptors (NMDARs) triggers LTP when followed by activation of D₁ dopamine receptors (D₁Rs), while calcium from voltage-gated L-type channels (Ca_v1.3) triggers LTD if paired with inactivation of D₁Rs (Shen et al., 2008; Yagishita et al., 2014; Fisher et al., 2017; Fino et al., 2010; Shindou et al., 2011; Iino et al., 2020). A threshold for eliciting LTD in SPNs has been experimentally quantified (Shindou et al., 2011). Although a similar threshold for LTP in SPNs has not been reported yet, presumably such a threshold exists, because weaker stimulation protocols do not induce LTP (Fino et al., 2005). (However, probably due to the local striatal microcircuitry, a range of different stimulation protocols can induce a range of different plasticity outcomes, as reviewed in Reynolds and Wickens (2002). An LTP threshold for calcium has been quantified in hippocampal CA1 neurons (Cormier et al., 2001).

By regulating when plasticity can be induced, metaplasticity has been proposed to prepare neural circuits and networks for learning, stabilize synaptic weights (prevent them from reaching very high or very low values) and to contribute to cognitive effects of disease when it is dysregulated (Hulme et al., 2013). All metaplasticity models so far have only one modifiable threshold, and typically, synaptic activity above the threshold induces LTP, while activity below the threshold induces LTD (Yger and Gilson, 2015; Bienenstock et al., 1982a; Benuskova and Abraham, 2006). Using a multicompartment model of an SPN, in this article we study what happens when there are two separate calcium thresholds for LTP and LTD, and what their role is in learning. We use a simple learning rule that models plasticity of the dSPN cortico-striatal synapse, to which we add metaplasticity in the two thresholds (Fig. 1A), and apply it to both the linear and nonlinear feature binding problem (FBP and NFBP). The results show that metaplasticity in both thresholds is necessary for successful learning of both tasks, with a specific, opposite and complementary role of each threshold.

Figure 1.

Before presenting the results, we introduce the FBP and NFBP. The FBP refers to combining (binding) an object’s individual features, such as color, shape or motion, into a unified object (Roskies, 1999; Treisman, 1999; von der Malsburg, 1999; Hardcastle, 2017). Feature binding is a problem because features such as color, shape and motion, are each processed in different brain regions, and possibly at different times, and need to be combined to represent/construct the perception of the object they correspond to. By applying the simple plasticity and metaplasticity rule to the FBP and NFBP, we in fact study whether a single SPN can perform feature binding, and focus on the role of metaplasticity in learning. In its simplest form, the linear feature binding problem (FBP) consists of discriminating between two patterns, meaning that a neuron should spike when activated by one of them, and remain silent when activated by the other (Fig. 1B, top). Each pattern consists of two features, e.g. shape and color, and one of the features can have two values (e.g. ‘red’ and ‘yellow’ for the color feature, as opposed to just ‘strawberry’ for the shape feature in Fig. 1B). Each feature excites the neuron equally, and the term “linear” refers to the computational complexity of the problem, meaning that a neuron able to perform simple linear summation (integration) of incoming stimuli, such as synaptic potentials summated (added) at the soma, can solve the problem. The nonlinear feature binding problem (NFBP) consists of discriminating between four patterns, so that the neuron spikes for two patterns and remains silent for the other two (Fig. 1B, bottom). Each pattern also consists of two features, and in the NFBP both features can have two values (‘strawberry’ or ‘banana’ for shape, and ‘red’ or ‘yellow’ for color, shown in the second table in Fig. 1B). Again, each feature excites the neuron equally, and the additional layer of complexity comes from the requirement to spike for two different patterns whose features do not overlap (‘red strawberry’ and ‘yellow banana’). One of the theoretical solutions to the NFBP requires supralinear integration of inputs (Tran-Van-Minh et al., 2015). SPNs exhibit such supralinearities in the form of prolonged dendritic voltage elevations called plateau potentials, demonstrated in the next section. (Supralinear integration of inputs can also solve the simpler FBP.)

As mentioned above, in addition to calcium influx, dopamine signals are also required for plasticity in SPNs. Because of this, we assume that after a pattern is presented, dopamine signals are emitted from the midbrain, and they indicate whether the pattern was “rewarding” or not (Fig. 1C). The patterns for which the SPN should learn to spike are called relevant patterns, and those for which it should be silent are called irrelevant patterns. We assume that dopamine peaks are emitted for relevant patterns, and dopamine pauses for irrelevant patterns (Fig. 1C). After stimulation of the synapse, if the required calcium signal has reached its corresponding threshold, the dopamine signal determines the direction of plasticity. Dopamine peaks are assumed to activate the D₁Rs, enabling LTP, and dopamine pauses are assumed to deplete dopamine enough so that D₁Rs are not activated, enabling LTD.

Results

Linear and supralinear integration of inputs by SPNs and corresponding calcium elevations

Figure 2 compares linear integration of synaptic inputs distributed across the dendrites with supra-linear integration of clustered synaptic inputs on one dendrite (Fig. 2A). The number of synaptic inputs is varied from 1 to 30 synapses, and the evoked somatic and spine voltage, as well as the spine [Ca]_NMDA and [Ca]_L-type are shown in Figs. 2B and 2C. A comparison of the amplitudes of these four quantities between linear and supralinear integration is shown in Fig. 2D. With linear integration, the somatic voltage amplitude increases linearly with the number of synapses (Fig. 2B₁ and blue line in Fig. 2D₁). Conversely, the spine voltage amplitude remains approximately the same, because spines are distributed across the dendrites and are activated in the same way (Fig. 2B₂ and blue line in Fig. 2D₂). The similar amplitudes of the spines’ voltage result in similar [Ca]_NMDA and [Ca]_L-type elevations (Figs. 2B₃, B₄ and blue lines in Figs. 2D₃, D₄).

Figure 2.

On the other hand, when synapses are clustered closely together on a dendrite, SPNs can perform supralinear integration of synaptic inputs through plateau potenitals (Du et al., 2017; Plotkin et al., 2011). Plateau potentials are evoked by activating synaptic NMDARs, and, in this study, in addition by glutamate spillover which activates extrasynaptic NMDARs. We included glutamate spillover because it robustly provides an all-or-none jump in the (somatic) voltage amplitude Trpevski et al. (2023). This all-or-none supralinearity is important for solving the NFBP (Tran-Van-Minh et al., 2015). Fig. 2C₁ shows the somatic voltage elevations when varying the size of the synaptic cluster from 1 to 30 synapses, while Fig. 2C₂ shows the spine voltage in one spine of the synaptic cluster. Similarly, Figs. 2C₃, C₄ show the spine [Ca]_NMDA and [Ca]_L-type that result from stimulating the synaptic cluster. As explained in the Methods section, calcium from these two different sources is tracked separately in the SPN model. Also, [Ca]_NMDA diffuses in the neuron, while [Ca]_L-type does not, since it is assumed to be localized in a microdomain. As the cluster size is increased, increasingly larger and longer voltage elevations are produced, as a result of activating more NMDARs (Figs. 2C₁, C₂, traces for cluster sizes below 20 synapses). Once glutamate spillover occurs (here at a cluster size of 20 synapses), an all-or-none plateau potential appears, causing an all-or-none jump in the calcium concentration, as well (Figs. 2C₁-C₄).

This all-or-none behavior is also visible as a jump in the amplitudes of the somatic and spine voltage, and the spine [Ca]_NMDA and [Ca]_L-type in Figs. 2D₁-D₄. The somatic and spine plateau amplitude shows roughly linear increases before and after glutamate spillover, with the largest non-linearity being caused by the plateau (Figs. 2D₁-D₂, red lines). The plateau also causes a jump in [Ca]_NMDA and [Ca]_L-type, as well (Figs. 2D₃-D₄, red lines). The jump in all four quantities in Fig. 2D is an indication of a “threshold supralinearity”. When a threshold level of stimulation is achieved, the neuron’s somatic voltage amplitude is suddenly much larger, despite it being stimulated with only one additional synapse in the synaptic cluster.

Also, the comparison of linear and supralinear integration in Fig. 2D₁ shows that, once a plateau appears, the same number of synapses trigger a much larger somatic voltage elevation when they are clustered compared to when they are randomly distributed across the dendrites. (It also shows a small difference between linear and supralinear integration when the number of stimulated synapses is less than 10. This arises because the synaptic clusters are always placed at approximately the same distance from the soma, while the distributed synapses can be randomly placed further away.)

Synaptic plasticity rule with metaplasticity solves both the FBP and NFBP

The synaptic plasticity rule is simple, and is given with Eq. 1. The synaptic weight w is increased if, during pattern presentation, [Ca]_NMDA reaches a level above the LTP threshold, θ_LTP, and a dopamine peak arrives afterwards. Similarly, the weight is decreased if [Ca]_L-type goes above the LTD threshold, θ_LTD, and a dopamine pause arrives afterwards. Synapses can be increased to a maximal level of w_max, and decreased to a minimal level of w_min, and weight updates are “multiplicative”, i.e. proportional to the current weight value. This formulation is very similar to existing plasticity rules (Rubin et al., 2001; Kistler, 2002; Gütig et al., 2003). The weight updates are also proportional to a learning rate η, and the time step for advancing the simulation, Δt. This rule is local, meaning that it depends on calcium levels locally at the synapse, and somatic spiking is not necessary for plasticity (similarly to Khodadadi et al. (2025)). This is beneficial for distal synapses, as backpropagating action potentials in general do not affect the local voltage which triggers calcium influx at distant dendritic locations. Also, at least in hippocampal CA3 pyramidal neurons, it has been reported that local dendritic depolarizations are necessary for plasticity, and somatic spiking is not, consistent with the fact that local calcium levels, driven by local voltage, are the signals triggering plasticity (Brandalise et al., 2016).

Metaplasticity is also implemented in a simple way, given with Eq. 2. It only occurs if synaptic plasticity also occurs. During one update step, each threshold moves closer to the amplitude (maximum) of the calcium level evoked by an arriving pattern, with a rate η_θ (θ_LTP follows the amplitude of [Ca]_NMDA, and θ_LTD follows the amplitude of [Ca]_L-type). This implementation is very similar to the sliding threshold in the well-studied BCM learning rule, where the threshold follows a quantity derived from local synaptic activity (Bienenstock et al., 1982a). When a synapse is modified, both thresholds are updated.

When using supralinear integration of inputs, we have added an upper threshold for LTP, Θ_LTP (Eq. 3). This threshold serves to stop synaptic strengthening once synapses are strong enough to evoke a plateau. With it, synapses are not strengthened to the maximal value, w_max, thus avoiding strong supralinear integration for the irrelevant patterns. (This is important for solving the NFBP, as will be seen below.) The upper threshold for LTP only plays a role in synaptic plasticity, and is not taken into account in metaplasticity. This means that metaplasticity is performed in the same way as in Eq. 2, and uner the conditions in Eq. 1.

In summary, each synapse is updated based on its local calcium levels, and has its own LTP and LTD thresholds. During learning, patterns in the FBP and NFBP are presented in random order, and after some time, each pattern is followed by a dopamine feedback signal (a dopamine peak or pause), during which plasticity occurs (Fig. 3A). The learning rule is always active (always “on”), avoiding separate training and testing phases sometimes used in other plasticity rules.

Figure 3.

Figure 3 shows the outcome of learning on the FBP and NFBP. For the FBP we have tested both linear integration of randomly distributed synaptic inputs (Figs. 3B₁-D₁) and supralinear integration of clustered synapses (Figs. 3B₂-D₂), while for the NFBP we tested only supralinear integration of clustered synapses in two different dendrites (Figs. 3B₃-D₃), since only supralinear integration can solve the NFBP (Tran-Van-Minh et al., 2015). Figs. 3C₁-C₃ show the voltage before and after learning for each pattern in the three scenarios. The SPN is silent before learning, and at the end of learning it has learned to spike to the relevant pattern(s). With supralinear integration, it does this by learning to evoke plateau potentials for the relevant pattern(s). For example, after learning (black traces), ‘red strawberry’ in Figs. 3C₂ and 3C₃ elicits a plateau in dendrite 1, which triggers somatic spiking. For the NFBP, the two relevant patterns evoke plateaus on separate dendrites (‘red strawberry’ on dendrite 1 and ‘yellow banana’ on dendrite 2, Fig. 3C₃). Figs. 3D₁, D₂ show the performance on the FBP with linear and supralinear integration averaged over 50 trials, indicating that both modes of integration can solve the task. For the NFBP, in addition to the input configuration in Fig. 3B₃, with three features on two dendrites, we also tested all possible configurations of four features on two dendrites which allow storage of the relevant patterns on separate dendrites. (These input configurations are listed in Figure 3–figure supplement 1.) Fig. 3D₃ shows that when up to three features are present in one dendrite, the NFBP is successfully solved. When four features are present in one or both dendrites, usually only one relevant pattern is learned. Solving the NFBP in this case requires additional mechanisms such as branch plasticity, as suggested by another study with more abstract neuron models and plasticity rules (Legenstein and Maass, 2011).

Before separately examining the effects of each calcium threshold on learning, it is first necessary to understand how the learning rule successfully solves the tasks by modifying the synaptic weights and thresholds. Figure 4 shows how the synaptic weights and the two calcium thresholds change during learning: they show the same qualitative behavior for both the FBP and NFBP. For the FBP, the synapses carrying the features ‘red’ and ‘strawberry’, which together form the relevant pattern, are strengthened (Figs. 4B₁, B₂). The synapses for ‘yellow’ only weaken, since they are only activated as part of the irrelevant pattern (which is followed by a dopamine pause, a signal for LTD). For the NFBP, where each relevant pattern is stored in a separate dendrite, the outcome is similar: the synapses for ‘red’ and ‘strawberry’ on dendrite 1 and the synapses for ‘yellow’ and ‘banana’ on dendrite 2 are strengthened (Figs. 4B₃, B₄). The remaining synaptic cluster, belonging to one irrelevant pattern in each dendrite, is weakened (the (the synapses for ‘yellow’ on dendrite 1 and the synapses for ‘strawberry’ on dendrite 2). In all cases, weakening stops when the evoked [Ca]_L-type falls below the LTD threshold. For the FBP with linear integration at the soma, many of the randomly distributed synapses for the relevant pattern are strengthened almost to the maximal value w_max (Fig. 4B₁). On the other hand, for the FBP with supralinear integration and the NFBP, the clustered synapses for the relevant pattern(s) are not strengthened to their maximal values. Instead, they stabilize once they are strong enough to evoke a plateau (Figs. 4B₂-B₄). This is due to the upper threshold for LTP, Θ_LTP, shown with dashed lines in Figs. 4C₂-C₄. (The stabilization takes about twice as long in the NFBP simply because there are twice as many patterns in the task.) When plateaus are evoked by the relevant patterns, they drive somatic spiking (as indicated above in Figs. 3C₂, C₃). An irrelevant pattern stimulates only half of the strengthened synapses on one dendrite. This is not enough for a plateau potential, and even though a supralinear voltage elevation is elicited, no somatic spiking is triggered (Fig. 3C₂, ‘yellow strawberry’ on dendrite 1 for the FBP and Fig. 3C₃, ‘yellow strawberry’ on dendrite 1 and ‘red banana’ on dendrite 2 for the NFBP). Nevertheless, these voltage elevations produce high calcium of both types, but, as will be seen below, the synapses involved are protected from weakening as a result of metaplasticity.

Figure 4.

The LTP and LTD thresholds for just one synapse from each feature are shown in Figs. 4C and 4D, respectively (the same plots are shown with larger panels in Figure 4–figure supplement 1 for better clarity). As mentioned above, they show the same behavior for both the FBP and NFBP, the only difference being that they stabilize at different values due to the different calcium levels evoked by distributed and clustered synaptic inputs. Also, the LTP and LTD thresholds show the same behavior when compared to each other. For example, looking at the strengthened synapses first, they increase to follow the calcium amplitudes evoked by each pattern presentation, and stabilize at these values towards the end of the simulation (thresholds for ‘red’ and ‘strawberry’ tend to the dots of the same color above them in Figs. 4C₁-C₃ and 4D₁-D₃, and the thresholds for ‘yellow’ and ‘banana’ tend to the dots of the same color above them in Figs. 4C₄, D₄; this is better visible in Figure 4–figure supplement 1 and also highlighted with the light blue regions in Figure 4–figure supplement 1B₃). In the case of linear integration they stabilize at low values, because distributed synapses do not have cooperative voltage effects (due to the large distances between them), and low calcium levels are evoked as a result (Figs. 4C₁, D₁). Instead, with supralinear integration, the plateaus evoked by the clustered synapses cause much larger voltage and calcium elevations, so the thresholds rise to much higher levels (Figs. 4C₂-C₄ and 4D₂-D₄).

We next look at the thresholds for the weakened synapses (‘yellow’ synapse in Figs. 4C₁-C₃ and 4D₁-D₃ and ‘strawberry’ synapse in Figs. 4C₄, D₄). At the beginning, the thresholds increase. Later, once the synapses are weakened enough so that the evoked [Ca]_L-type falls below the LTD threshold, they reach a stable level. For example, in Figs. 4C₂, C₃ and 4D₂, D₃, at the beginning of learning, yellow dots can be seen above the ‘yellow’ thresholds, while later they almost always remain below the thresholds. Similarly, in Figs. 4C₄, D₄, light red dots are at first seen above the ‘strawberry’ thresholds, while later they are found below them. are at first seen above the ‘strawberry’ thresholds, while later they are found below them. This transition of the [Ca]_L-type amplitudes from above to below the LTD thresholds is marked with the arrows in Figs. 4B_2-4-D_2-4, corresponding to the moment in time where synapses are weakened enough. (This is more clearly visible in Figure 4–figure supplement 1, and it is also highlighted with the green and brown regions in Figure 4–figure supplement 1B₃, and the pink region where yellow dots are absent.) We mention that in the FBP, where the weakened synapses are only activated by the irrelevant pattern, both of their thresholds are only updated during LTD. On the other hand, in the NBFP the weakened synapses and their thresholds can be updated during both LTP and LTD, because these synapses are also activated by a relevant pattern. (However, in Fig. 4 this only happens at the very beginning, when the weakened synapses also undergo LTP. As they are being weakened, their LTP threshold is raised, so [Ca]_NMDA amplitudes later do not go above the LTP threshold.) In summary, once a calcium amplitude falls below its threshold, one of the necessary conditions to trigger plasticity and metaplasticity is not met, and the thresholds, as well as the synapses, are not updated anymore. This will be much more evident in the next section, where we study the role of metaplasticity.

We stress once again that with supralinear integration, one of the strengthened synaptic clusters in each dendrite is activated by both a relevant and an irrelevant pattern (the ‘strawberry’ cluster in Figs. 4B₂, B₃ and the ‘yellow’ cluster in Fig. 4B₄). Importantly, the LTD threshold for these synapses is raised because of the plateaus (Figs. 4D₂-D₄: ‘strawberry’ synapse in Figs. 4D₂, D₃ and ‘yellow’ synapse in Fig. 4D₄). Because of this, the high [Ca]_L-type levels in these synapses evoked from an irrelevant pattern are below the LTD threshold and are not enough to weaken them (e.g. light red dots in Figs. 4D₂, D₃ just above the ‘yellow’ threshold after 100 s are far below their ‘strawberry’ threshold; similarly yellow dots in Fig. 4D₄ located just above the ‘strawberry’ threshold after 100s are far below their ‘yellow’ threshold; dots also highlighted in the pink region of Figure 4–figure supplement 1B₃). In this way, these strengthened synapses are protected from weakening, allowing them to store one part of the relevant pattern. This effect will also be much more evident in the next sections, where we look at the effects of metaplasticity on learning.

Briefly, we have also examined the effects of parameters in the learning rule. In supralinear integration, the upper plasticity threshold Θ_LTP prevents the strengthened synapses to saturate at their maximal levels. Without it, the FBP is still solved, even though weights attain their maximal values (Figure 4–figure supplements 2B₁ and 2E₁). It is successfully solved because the features innervate only one dendrite, and the saturated weights do not cause somatic spiking when activated by the irrelevant pattern (Figure 4–figure supplement 2A₁). However, the NFBP is not solved without an upper threshold Θ_LTP, because the features innervate two dendrites, and the maximally strengthened synapses cause supralinear voltage elevations in the dendrites for the irrelevant patterns, which trigger somatic spiking (Figure 4–figure supplements 2A₂, 2B₂, 2B₃ and 2E₂). The maximally strengthened synapses also drive more weakening in the weakened synapses, compared to when an upper threshold is present (Figure 4–figure supplement 2B). This is due to the elevated [Ca]_L-type evoked when an irrelevant pattern activates the maximally strengthened synapses (‘strawberry synapses in Figure 4–figure supplements 2B₁, B₂ and ‘yellow’ synapses in Figure 4–figure supplement 2B₃). The elevated [Ca]_L-type for the irrelevant patterns is also visible in the elevated LTD thresholds for the weakened synapses (‘yellow’ synapse in Figure 4–figure supplement 2D₁, D₂ and ‘strawberry’ synapse in Figure 4–figure supplement 2D₃, cf. Figs. 4D₂-D₄). (The LTP thresholds for the weakened synapses are also elevated, due to higher [Ca]_NMDA produced by the maximally strengthened synapses. This is visible in Figure 4–figure supplement 2C when compared to Figs. 4C₂-C₄)

The learning rate η has an effect on the speed of learning - more pattern presentations are required with lower learning rates, and vice versa (Figure 4–figure supplement 3). The rate of threshold adaptation (metaplasticity rate) η_θ has no significant effect on learning to solve the FBP and NFBP (Figure 4–figure supplement 4).

Metaplasticity in the LTD threshold prevents weakening of strengthened synapses, protecting learned patterns

The role of each threshold is easily examined with this plasticity and metaplasticity learning rule by turning metaplasticity off, i.e. fixing a threshold’s value during learning. We focus on the NFBP because it is the harder task, but we also check metaplasticity’s effect on the FBP. Fixing the LTD threshold (to the initial low value in the simulation) shows that the synapses encoding the relevant patterns cannot strengthen properly (Figs. 5C₁, C₂). Before we show why this is so, it is important to remember that in the NFBP, a relevant pattern shares one feature with both irrelevant patterns (e.g. ‘red strawberry’ shares the color feature with ‘red banana’ and the shape feature with ‘yellow strawberry’). Because of this, half of the synapses activated by a relevant pattern are also always activated when an irrelevant pattern arrives.

Figure 5.

As learning progresses, the relevant patterns cause strengthening in the synapses that they activate. However, because there is no metaplasticity in the LTD threshold, irrelevant patterns always cause higher [Ca]_L-type than the fixed LTD threshold (Figs. 5E₁, E₂ – LTD thresholds stay fixed during learning, and [Ca]_L-type is always above θ_LTD). This means that when the strengthened synapses are activated by an irrelevant pattern, they will always get weakened. Conversely, when metaplasticity in the LTD threshold is active, it follows the increased levels of [Ca]_L-type elicited by a plateau (evoked due to previous synaptic strengthening, as shown previously in Figs. 4D₂–D₄). This elevated LTD threshold protects the strengthened synapses from weakening when they are activated by an irrelevant pattern. As mentioned above, they are protected because an irrelevant pattern, activating only half of the strengthened synapses, does not evoke a plateau, resulting in [Ca]_L-type lower than the elevated LTD threshold in Figs. 4D₂ - 4D₄ (e.g. light red dots in pink region in Figure 4–figure supplement 1B₃ are below the elevated ‘strawberry’ threshold).

Without metaplasticity in the LTD threshold, strengthened synapses are not protected from weakening. As a result, the synaptic weights for the relevant pattern cannot stabilize (Figs. 5C₁, 5C₂), and no plateaus are evoked to trigger somatic spiking (Fig. 5B), so the NFBP cannot be learned (Fig. 5F). The LTP threshold follows the level of [Ca]_NMDA and also does not stabilize (Figs. 5D₁, 5D₂). Also, because the LTD threshold is fixed to a low value, the synapses belonging to the weakened feature in each dendrite are almost weakened to the minimal value w_min. In summary, by protecting synapses from weakening, the threshold for weakening allows proper synaptic strengthening.

In the FBP, metaplasticity in the LTD threshold has the same effect. However, due to there being only two patterns in the FBP, the synapses for only one feature in the relevant pattern cannot stabilize (the ‘strawberry’ synapses, Figure 5–figure supplement 1B). This is because the synapses for the feature ‘red’ are only activated by the relevant pattern and, as a result, can only ever be strengthened. Phrased differently, they are never activated by an irrelevant pattern, which would cause them to weaken and be influenced by the fixed LTD threshold. Because the ‘strawberry’ synapses do not stabilize, the clustered synapses in the FBP with supralinear integration are not strong enough to evoke plateaus, and the FBP is not learned (Figure 5–figure supplements 1A₂ and 1E₂). On the other hand, in the FBP with linear integration, despite these synapses not stabilizing, the strengthening is enough to drive somatic spiking (Figure 5–figure supplement 1A₁). The FBP with linear integration is solved but with a lower performance (Figure 5–figure supplement 1E₁).

Metaplasticity in the LTP threshold prevents strengthening of weakened synapses, promoting the weakening of unnecessary synapses

On the other hand, the LTP threshold has an opposite effect on learning. If this threshold is fixed, the feature belonging to the irrelevant pattern in each dendrite cannot be weakened properly (Figs. 6C₁, C₂). Because of the low value of the LTP threshold, whenever an irrelevant pattern activates these synapses, the evoked [Ca]_NMDA is still greater than the threshold (Figs. 6D₁, D₂), triggering synaptic strengthening even if synapses have been previously weakened. This means that when metaplasticity in the LTP threshold is active, by following the levels of [Ca]_NMDA (e.g. LTP threshold for ‘yellow’ in Figs. 4C₁, C₂, and for ‘strawberry’ in Fig. 4C₃), it prevents these synapses from strengthening when they are activated by the relevant pattern (which can only be stored fully in the other dendrite). As a result, the synapses for ‘yellow’ in Figs. 4B₁, B₂, and for ‘strawberry’ in Fig. 4B₃ are successfully weakened, while in Figs. 6C₁, C₂ they are not. (The LTD threshold behaves as usual for the strengthened synapses, protecting them from weakening, as shown in Figs. 6C₁, C₂ and 6E₁, E₂. However, for the weakened synapses, it does not stabilize.)

Figure 6.

Without metaplasticity in the LTP threshold, the NFBP is on average not solved (Fig. 6F). Despite both relevant patterns being stored by each dendrite, the performance is lowered because the SPN sometimes also spikes for the irrelevant patterns (Fig. 6B). Also, in the input configurations containing four features in one dendrite, the average performance even drops below 50 % (green trace in Fig. 6F). In summary, by preventing synapses from strengthening, the threshold for strengthening allows proper synaptic weakening.

In the FBP, a fixed LTP threshold does not have an effect on learning (Figure 6–figure supplement 1). The synaptic weights behave the same as with metaplasticity in the LTP threshold being active (Figure 6–figure supplement 1B, cf. Figs. 4B₁, B₂). As a result, the FBP is solved with both linear and supralinear integration (Figure 6–figure supplements 1A and 1E). This is also due to there being only two patterns in the FBP. The weakened ‘yellow’ synapses, whose proper weakening could be affected by the fixed LTP threshold, are only activated during the irrelevant pattern (‘yellow strawberry’). Because they are never activated by the relevant pattern, they can never strengthen, and as a result they remain unaffected by the fixed LTP threshold.

Both thresholds need to be updated during one synaptic modification in order to solve the NFBP

The rule is formulated so that during one synaptic modification (e.g. LTP) the thresholds for both LTP and LTD are updated. Alternatively, it could be formulated so that during LTP, only the threshold for LTP is updated, and during LTD, only that for LTD is updated. We test this variant in this section, and the results from the previous two sections already give an intuition how this variant would work. Updating only the corresponding threshold during one synaptic modification means that the other threshold is, in effect, kept fixed during that plasticity update, making metaplasticity only partially active. The effects are seen in Figs. 7C₁, C₂. Synapses are neither properly strength-ened, nor weakened, at least within the length of the simulation. For example, one of the relevant patterns (‘red strawberry’) shows a trend of strengthening, but is also being weakened (Fig. 7C₁). This is because as the LTD thresholds for ‘red’ and ‘strawberry’ move higher (Fig. 7E₁), LTD in these synapses is (sometimes) prevented, and a trend of synaptic strengthening appears. Also, as the LTP threshold for ‘yellow’ increases (Fig. 7D₁), it prevents these synapses from strengthening, and as a result, they weaken to a degree.

Figure 7.

However, since the LTD threshold is not updated during LTP, it does not follow the [Ca]_L-type levels generated by the relevant patterns. It is updated only during LTD, following the [Ca]_L-type levels from irrelevant patterns. For example, in dendrite 1, the ‘red banana’ activates only the cluster representing ‘red’, evoking little [Ca]_L-type and, as a result, the LTD threshold for the ‘red’ synapse is lower (Fig. 7E₁). On the other hand, the ‘yellow strawberry’ activates two synaptic clusters (Fig. 7A₁), causing a larger voltage elevation and consequently larger [Ca]_L-type, so the LTD thresholds for these synapses move higher (Fig. 7E₁). However, because of the trend of synaptic strengthening in the ‘red’ and ‘strawberry’ synapses, the irrelevant patterns sometimes still evoke [Ca]_L-type higher than the LTD thresholds. As a result, the synapses for ‘red’ and ‘strawberry’ in dendrite 1 are not protected from weakening (when activated as part of the irrelevant patterns). Also, as the LTD threshold for ‘yellow’ increases in the first 70 s of the simulation (Fig. 7E₁), the synapses for ‘yellow’ show a trend of increasing in this period, as well, before being decreased afterwards (when the ‘yellow’ LTP threshold is increased).

On the other hand, in dendrite 2 the relevant pattern is not strengthened, and it is not clear whether more learning trials will cause both relevant patterns to be stored in the two dendrites. On average, the NFBP is not solved in this case (Fig. 7F).

Finally, this variant is similar to not having metaplasticity at all (Fig. 8). Without metaplasticity, no synapses stabilize, i.e. they are neither strengthened nor weakened (Fig. 8C₁, C₂).

Figure 8.

For the FBP, the results in the previous two sections again give an intuition for how partial meta-plasticity would work. As shown in Figure 6–figure supplement 1, even when the LTP threshold is fixed, it does not have an effect on solving the FBP. Similarly, neither does it have an effect when it is partially active: the LTP threshold is merely less elevated than compared to when metaplasticity is always active (Figure 7–figure supplement 1C, cf. Figs. 4C₁, C₂). On the other hand, partial meta-plasticity in the LTD threshold is not enough to protect strengthened synapses from weakening. The LTD threshold does not follow [Ca]_L-type fast enough (Figure 7–figure supplement 1D), and as a result the ‘strawberry’ synapses are not strengthened properly (even though, for example, their weights stabilize when using supralinear integration, Figure 7–figure supplement 1B). Similarly to Figure 5–figure supplement 1, this does not affect learning to solve the FBP with linear integration (Figure 7–figure supplement 1A₁ and 1E₁), but it does affect learning of the FBP with supralinear integration (Figure 7–figure supplement 1A₂ and 1E₂).

The effects of having no metaplaticity on the FBP are similar to those of having partial meta-plasticity (Figure 8–figure supplement 1). Again, the synapses encoding one of the features in the relevant pattern cannot stabilize (those for ‘strawberry’, Figure 8–figure supplement 1B). This only moderately affects the performance on the FBP with linear integration (Figure 8–figure supplement 1A₁ and 1E₁), but is necessary for solving the FBP with supralinear integration (Figure 8–figure supplement 1A₂ and 1E₂). In summary, this shows that in the FBP, metaplasticity is important when a feature is represented with clustered synapses. It allows their synaptic weights to stabilize, and to evoke plateau potentials that can drive somatic spiking. Conversely, when a feature activates distributed synapses across the dendrites, the FBP can still be solved without metaplasticity, even if some synapses do not stabilize.

Discussion

In this article we studied the effects of metaplasticity in two separate calcium thresholds for inducing synaptic plasticity, one of which is a threshold for LTP, and the other for LTD. Plasticity occurs when calcium levels from different sources, implicated in synaptic plasticity in SPNs, reach levels above the thresholds. We studied the effects of metaplasticity on how well the SPN learns the FBP and NFBP, tasks which require linear and supralinear summation of synaptic inputs, respectively. The results showed that:

1. Without metaplasticity, the NFBP cannot be solved. It is also necessary for solving the FBP with supralinear integration, but it is not necessary for solving the FBP with linear integration.

2. Metaplasticity in the LTD threshold protects learned patterns from being weakened, allowing the LTP process to properly strengthen (encode) the patterns. In effect, the threshold for weakening allows proper synaptic strengthening.

3. Metaplasticity in the LTP threshold prevents unnecessary features from being strengthened, allowing the LTD process to persistently weaken them. In effect, the threshold for strengthening allows proper synaptic weakening.

4. For successful learning of the NFBP, metaplasticity in both thresholds needs to occur during one synaptic modification, irrespective of whether the modification is LTP or LTD.

In summary, the two calcium thresholds have opposite and complementary roles in regulating the strengthening and weakening of synapses. The ultimate effects of metaplasticity are unexpected, at least at first glance, since the threshold for one plasticity type (e.g. LTD) allows proper expression of the other plasticity type (e.g. LTP). Metaplasticity is particularly important for clustered synapses, where the cooperative, supralinear voltage elevations can cause large calcium influx - it allows synpatic plasticity to adapt to this large calcium influx. By regulating plasticity in clustered synapses, metaplasticity allows neurons to perform computationally difficult tasks (such as the NFBP) that rely on supralinear integration by the clustered synapses. Simpler tasks that can be solved with linear integration of synaptic inputs (such as the FBP) do not require metaplasticity.

This study also provides a basis for exploring other metaplasticity rules. Here we only considered metaplasticity that is activated as a consequence of synaptic plasticity. However, stimulation that does not cause synaptic plasticity can also activate metaplasticity, affecting the induction of future plasticity (Abraham, 2008). Also, we only considered metaplasticity evoked in the same synapses activated by presynaptic spikes (homosynaptic metaplasticity). However, heterosynaptic metaplasticity mechanisms also exist, where the evoking stimuli originate from different synapses or different neurons (Abraham, 2008; Abraham et al., 2019; Hulme et al., 2014). Furthermore, as detailed calcium models for other neurons are developed, the contributions of different calcium sources in triggering synaptic plasticity can be studied using this simple learning rule as a basis. Regarding SPNs specifically, more realistic metaplasticity rules can also be developed once future experimental findings appear.

We have made two assumptions when formulating the learning rule. The first is that metaplasticity operates in cortico-striatal synapses, which is not a strong assumption, since metaplasticity has been reported for many synapses. The second – a major assumption – is the upper threshold for synaptic strengthening, which means that high [Ca]_NMDA during LTP conditions (a dopamine peak) prevents synaptic plasticity (blocks LTP). It is an open question whether this assumption is true. Nevertheless, there are some molecular mechanisms which could form the basis for such an upper threshold. In the neuromodulatory branch of the signaling network in Fig. 1 (the branch activated by dopamine), the enzyme adenylyl cyclase 5, which transduces the dopamine signal from the D₁R, is inhibited by calcium. This indicates that high [Ca]_NMDA might prevent synaptic strengthening already very early in the signaling network. Further downstream in the signaling network, calcium activates PP2B through calmodulin, and PP2B inactivates PKA, an enzyme necessary for synaptic strengthening (Church et al., 2021; Yagishita et al., 2014). This is another route through which high [Ca]_NMDA could prevent LTP. Due to the complexity of the signaling network, the influence of high [Ca]_NMDA on synaptic plasticity requires further study, and kinetic models of the cortico-striatal synaptic signaling network which incorporate the many molecular interactions might provide some answers (Lindskog et al., 2006; Nakano et al., 2010; Nair et al., 2015).

In summary, at least when it comes to solving the NFBP, the proposed learning rule highlights the importance of deciding when to stop updating the synaptic weights. The two processes (mechanisms) that regulate when synaptic plasticity occurs are the upper plasticity threshold for LTP and metaplasticity. Without the upper plasticity threshold, all strengthened synapses would reach their maximally attainable synaptic weight. Without metaplasticity, the synaptic weights would not stabilize. Important future questions relate to the various such possible mechanisms and their effects on learning, as it is likely that different types of synapses possess different metaplasticity mechanisms, which support specific functions in various brain regions. Some examples of different metaplasticity mechanisms are known in the hippocampus. In the Schaffer collateral synapses to the CA1 area, prior activation of NMDARs (which does not induce long-term plasticity) temporarily increases the LTP threshold and prevents LTP, while prior activation of mGluRs facilitates LTP (Huang et al., 1992; Cohen and Abraham, 1996). In the synapses from the perforant path to the dentate gyrus, prior stimulation at theta frequency typical for the hippocampus, lowers the threshold for both LTP and LTD (Christie et al., 1995; Christie and Abraham, 1992). Additionally, prior stimulation at a high frequency also lowers the LTD threshold in both unchanged and in previously strengthened synapses (Holland and Wagner, 1998).

Finally, we have recently participated in another study asking whether SPNs could solve the NFBP, using a different learning rule that incorporates more biological complexity/detail (Khodadadi et al., 2025). That study focused on dendritic integration which was much less supralinear, i.e. the dendrites were in a regime where they produced so-called “boosting” nonlinearities, that are likely common physiologically. Completely solving the NFBP required both excitatory and inhibitory synaptic plasticity, regardless of whether the synaptic clusters were closer or further away from the soma. On the other hand, the all-or-none plateau potentials in this study achieve a very high score on the NFBP with excitatory plasticity alone. This indicates that with a threshold non-linearity, the SPNs might completely solve the task without requiring inhibitory synapses and even for very distally located synaptic clusters. Whether this is true will be explored in another study, as theoretical considerations suggest that the NFBP can be solved with excitatory synapses only (Tran-Van-Minh et al., 2015).

Methods

Plasticity rule

We highlight again that, similar to Khodadadi et al. (2025), the learning rule is:

• local: only signals at the local synaptic site are used to trigger plasticity, calcium- and reward-based:

• local calcium levels trigger plasticity, the type of which is decided by the dopamine reward signal, and

• always “on”: no separate training and testing phases are needed because learning is regulated by metaplasticity.

Differently from Khodadadi et al. (2025), the learning rule has metaplasticity in both calcium thresholds, and is also (much) simpler, making it suitable for studying the thresholds’ roles in synaptic plasticity.

The parameters used in the plasticity rule are given in Table 1. The values of w_max and w_min are not varied throughout the simulations. They describe the maximal and minimal values that the synaptic weight can take on in the model. Whether these values are reached depends on the metaplasticity mechanisms (the values of the calcium thresholds), and, in the case of supralinear integration, the upper plasticity threshold Θ_LTP, which is a fixed parameter that can be set at any value in the region of the supralinearity (the jump) in [Ca]_NMDA in Fig. 2D₃ (we used 40 μM). The learning rate η and the metaplasticity rate η_θ are also kept fixed throughout all simulations, except in Figure 4–figure supplements 3 and 4, where they are varied to study their effect on learning. Finally, the LTP and LTD thresholds, θ_LTP and θ_LTD, are initialized to low values, θ_{LTP, 0} and θ_{LTD, 0}, which are also the minimal values that these thresholds can take on. They are initialized to low values in order to showcase learning in the synapses. Otherwise, if they are set to high values, they may act to protect synapses from weakening (high θ_{LTD, 0}) or prevent them from strengthening (high θ_{LTP, 0}), as is shown with Figs. 5 and 6. The synaptic weights are initialized uniformly at random from the interval [0.3, 0.35].

Table 1.

Neuron model

We use a morphologically realistic, multicompartmental model of a direct-pathway striatal projection neuron from the collection of 71 such models in Lindroos and Hellgren Kotaleski (2020) (index 34 in the collection). As detailed in Lindroos and Hellgren Kotaleski (2020), the ionic conductances in all models in the collection have been selected with a search process to produce frequency-current (F-I) responses and dendritic calcium elevations arising from backpropagating action potentials that match experimental data, and in addition produced a suitable number of spikes to injected current, no voltage oscillations leading up to the spikes, proper afterhyperpolarization behavior, and an F-I slope within three standard deviations of the experimentally determined mean.

The main contributor to supralinear integration in the SPN model is the activation of NMDARs. This has been shown, for example, in Fig. S7 in Du et al. (2017), where supralinear integration with clustered synapses is similar in an active and a passive dendrite in the SPN. This means that the NMDA synapses, and not the SPN’s ion channels, are causing it. SPN–specific ion channels do have a large effect on the SPN’s elecrophysiological behavior – the lowered resting potential of around −85 mV is a result of the strongly inwardly rectifying K⁺ channels (Nisenbaum and Wilson, 1995; Wilson, 2005). Because of this, to raise their somatic voltage, the SPNs require strong synaptic drive from distributed synaptic inputs across the dendrites, or a localized activation of clustered synapses which generates a plateau potential. With this behavior, SPNs are said to have two states: a “down state”, which requires strong synaptic drive to reach the firing threshold, and an “up state”, where they are more sensitive to incoming synaptic inputs (Plotkin et al., 2011). It has also been suggested that plateau potentials themselves cause the transition from a “down” state to an “up” state (Oikonomou et al., 2014). Plateau potentials in SPNs normally do not cause somatic spiking without additional synaptic inputs, which is why in the simulations we have used elevated background noise.

In this study, synapses are placed on spines, which are not present in the original model in Lindroos and Hellgren Kotaleski (2020). The spines consist of a neck and head with neck and head with lengths and diameters l_neck = 0.5μm, l_head = 0.5μm, and dneck = 0.125μm, dhead = 0.5μm, respectively. Axial resistance in all compartments of the model is 150 Ω ⋅ cm, except for the spine neck, where it is 1130 Ω ⋅ cm (Dorman et al., 2018). The spines contain the inwardly rectifying K⁺ channel, and the same voltage-gated calcium channels as in the dendrites. The spine calcium channel conductances have been manually tuned to match the relative proportions determined in Carter and Sabatini (2004) and Higley and Sabatini (2010), as well as the Ca²⁺ concentration amplitudes arising from backpropagating action potential (bAP) stimulation as in Fig. 2 of Shindou et al. (2011).

Calcium dynamics and diffusion model

Because calcium from two different sources triggers LTP and LTD in SPNs, we tracked calcium from the two sources separately. Calcium from NMDA channels, when followed by a peak in dopamine, triggers LTP (Shen et al., 2008; Yagishita et al., 2014; Fino et al., 2010; Fisher et al., 2017). Calcium from Ca_v1.3 channels, when paired with a pause in dopamine, has been shown to be necessary for LTD (Wang et al., 2006; Fino et al., 2010; Shindou et al., 2011; Iino et al., 2020).

Without implementing calcium diffusion explicitly, the accumulation of [Ca]_NMDA in a spine shows a strange effect when the synaptic cluster size is gradually increased, which is that the largest plateaus do not correspond to the largest calcium amplitude (Figs. 9A, B). This is simply a result of less NMDAR membrane current flowing in as the plateau’s voltage approaches the NMDAR reversal potential in Eq. 6. Larger plateaus giving rise to lower [Ca]_NMDA amplitudes could pose a problem to the learning rule, since the decision when to trigger plasticity and metaplasticity is made by comparing [Ca]_NMDA amplitude against the LTP threshold. In a synaptic cluster that is being strengthened, but starts to evoke a smaller [Ca]_NMDA amplitude after some strengthening due to the effect in Figs. 9A, B, learning might be prematurely stopped as the [Ca]_NMDA amplitude falls below the increased LTP threshold.

Figure 9.

Because of this, we sought to obtain a monotonic increase of calcium amplitude with increasing plateau voltage (as in Fig. 2C₃), for which it was necessary to implement axial diffusion and buffering for [Ca]_NMDA. Because tuning a detailed calcium diffusion model to data is a project in its own right, we used the most detailed calcium diffusion model for SPNs available to date (by Dorman et al. (2018)) as a reference. We reuse the same buffer molecules, their amounts and their calcium binding and unbinding rates from Dorman et al. (2018) with small changes, given in Table 2. The rationale for reusing them is that the idealized morphology used in Dorman et al. (2018) should on average represent any other SPN, so the parameters obtained from the optimization procedure in that study should on average also apply to other multicompartment SPN models. The model in Dorman et al. (2018) also includes a plasma-membrane Ca pump (PMCA) in the soma, dendrites and spines, which we added in this model, and a sodium/calcium exchanger in the spines, which we ommited, because with the parameters from Dorman et al. (2018), it did not noticeably alter the spine [Ca]_NMDA concentration. In the calcium diffusion model in Dorman et al. (2018), calcium originating from all sources, which in that study are NMDARs and voltage–gated calcium channels (VGCCs), and a minor, 0.1% contribution of the AMPAR current, diffuses throughout the SPN model. Because the calcium buffer parameters are optimized to such a scenario, we also included calcium from the Ca_v2.1 (Q-type), Ca_v2.2 (N-type), Ca_v2.3 (R-type), Ca_v3.2 and Ca_v3.3 (T-type) channels in the SPN together with [Ca]_NMDA in the diffusion model. Indeed, non-selective blockade of voltage-gated calcium channels with mibefradil has shown that they are also necessary for inducing LTP in SPNs (Fino et al., 2010). Mibefradil has been shown to block T-type, L-type, R-type, N-type and P/Q-type calcium channels in different experimental preparations, so, apart from the L-type channels, we included all of them in the diffusion model (Randall and Tsien, 1997; Viana et al., 1997; Aczél et al., 1998; Eller et al., 2000; Leuranguer et al., 2001). What the role of each of these channels is in LTP in SPNs requires further study with selective blockers.

Table 2.

We also show how diffusion of only [Ca]_NMDA looks like in the SPN in Figure 9–figure supplement 1 (and Fig. 9C). The difference with the diffusion model used throughout the study, which includes VGCCs, is minor (Figure 9–figure supplement 1, cf. Figs. 2C, D). The qualitative and quantitative shapes of the voltage and calcium elevations are very similar (Figure 9–figure supplement 1A, cf. Fig. 2C). Regarding the amplitudes of these quantities, there is a 1 mV increase in the evoked somatic amplitude after glutamate spillover when only [Ca]_NMDA diffuses (Figure 9–figure supplement 1B₁), and an obvious difference in the amplitude of [Ca]_NMDA (Figure 9–figure supplement 1B₃). Nevertheless, the threshold nonlinearity in the [Ca]_NMDA amplitude is still present, and the learning rule would work the same provided the upper plasticity threshold Θ_LTP is placed in the region of the nonlinearity.

Even though mibefradil also affects L-type channels, we did not include them in the axial diffusion model together with [Ca]_NMDA because L-type calcium channels frequently form calcium microdomains, where calcium entry and diffusion is restricted within a very small, micrometer range (Berridge, 2006; Chen and Sabatini, 2012; Parekh, 2008). In the microdomain, the L-type channels can be physically coupled in supramolecular protein complexes, further contributing to the local effect of the calcium influx in activating the relevant signaling networks, and this is also the case for Ca_v1.3 channels in SPNs (Olson et al., 2005; Stanika et al., 2015). Because of this, we assume that such highly localized [Ca]_L-type signals trigger LTD in the synapse, and calculated [Ca]_L-type with a separate, phenomenological pool model, which describes [Ca]_L-type in a thin cylindrical shell under the membrane. The implementation follows that of Wolf et al. (2005), according to Eq. 4. In this model [Ca]_L-type decays with a time constant τ_Ca, modeling Ca²⁺ diffusion only phenomenologically, i.e no Ca²⁺ ions move between dendritic compartments. Compared to the very simplest pool models, this pool model also includes a Ca²⁺ pump that extrudes intracellular Ca²⁺ to the extracellular space. F is the Faraday constant, k_t is the catalytic activity of the pump, K_d is the pump dissociation constant for Ca²⁺, d is the depth of the submembrane shell where Ca²⁺ accumulates, and k and p are phenomenological parameters used in Wolf et al. (2005) used to balance Ca²⁺ influx and efflux. The values for the parameters in Eq. 4 are given in Table 3.

Table 3.

In this pool model we included calcium from both Ca_v1.2 and Ca_v1.3 channels, since there are no pharmacological agents that selectively block only one of the channels, i.e. the compounds used in the experiments which have determined Ca_v1.3 as necessary for LTD in cortico-striatal synapses would also block Ca_v1.2 channels in SPNs (Berger and Bartsch (2014); nimodipine in Wang et al. (2006) and Shindou et al. (2011); also, other non-selective L-type channel blockers are used to block LTD in the following studies: nifedipine in Calabresi et al. (1994); nitrendipine in Kreitzer and Malenka (2005); mibefradil in Fino et al. (2010); finally, experiments in Plotkin et al. (2013) have suggested that other VGCCs than Ca_v1.3 may be involved in activating calcium-induced calcium release, which is necessary for LTD).

Synaptic inputs

There are two different models of synapses used in this study. One model – the dual exponential model – is used in describing the background noise, which consists of glutamatergic and GABAergic synapses, distributed across the dendrites according to the distribution measured in Cheng et al. (1997). These synapses are fixed (they have no plasticity) because they are activated at a low rate of 1 Hz. Compared to the 35 ms activation window in Fig. 3, this rate is low enough so that we decided to neglect any background noise synapses that might be co-active in the same time window and thus become eligible for plasticity together with the feature-carrying synapses. Because there are many synapses representing the background noise, this assumption also lowers the computational cost – if they were plastic, the learning rule would need to be computed for each of them.

The other model, used to describe the synapses carrying the features, is a saturating synapse model. We chose a saturating synapse model because we activate these synapses with 3 spikes each within a 35 ms window, and wanted to avoid any unrealistically high conductance values that could otherwise arise in non-saturating models.

Dual exponential model

The dual exponential synaptic model (a difference of two exponential functions) for a spike arriving at time 0 is represented by:

where g_max is the maximal synaptic conductance, and τ_A and τ_B are the rise and decay time constants. C is a normalization factor given with:

where t^⋆ is the position of the maximum of the dual exponential function. This normalization ensures that the time-varying part of the dual exponential reaches a peak value of 1, which is then scaled by the maximal conductance g_max. (The implementation in NEURON contains a synaptic weight parameter, as well. Since in the dual exponential model we have fixed the weight to 1, it is omitted from Eq. 5.)

The synaptic current arising from the conductance g_syn is:

Excitatory synapses have both AMPA and NMDA components, and NMDA synapses in addition have a Mg²⁺ block described by a sigmoidal function:

with parameters α = 0.062 and η = 0.38. The parameters of the dual exponential model for the glutamatergic AMPA and NMDA synapses, and the GABAergic synapses are given in Table 4.

Table 4.

Finally, because the SPN model contains less electrical compartments than the number of synapses reported in Cheng et al. (1997), this implies that several background noise synapses will arrive in a single compartment. As such, their synaptic inputs will be integrated in the same voltage variable. Because of this, we have represented such multiple synapses arriving in one compartment with a single synapse instead, whose input frequency is scaled by the number of synapses it represents. This is why a non-saturating synapse model, such as the dual exponential model, was necessary for the background noise synapses. This also greatly increases simulation speed.

Saturating synapse model

The saturating synapse models are taken from Gao et al. (2021), which are a variation of the saturating synapse models in Destexhe et al. (1994). They are kinetic models which operate according to two different kinetic schemes depending on the presence or absence of neurotransmitters. When neurotransmitters arrive at the postsynaptic site, receptor dynamics are described by a kinetic scheme that models switching between closed (C) and open (O) states with rates of α and β:

where [T] is the transmitter concentration. After neurotransmitter has been cleared from the synaptic cleft, receptor dynamics evolves according to a kinetic scheme that describes just the closing of the receptor with a rate β:

In this model, when a presynaptic spike arrives, neurotransmitter levels always reach a fixed saturating concentration in the synaptic cleft, [T]_max, which lasts for a short time T_dur (a short glutamate lifetime in the synaptic cleft of cultured hippocampal synapses has been reported in Clements et al. (1992)). If another presynaptic spike arrives while the neurotransmitter pulse is still “on”, the pulse duration is lengthened by T_dur.

The synaptic conductance g_syn in this model is represented by the fraction of glutamate receptors in the open state, [O]:

where g_max is again the maximal synaptic conductance, and w is the synaptic weight which is modified by the learning rule. The parameters in the saturating synapse model are given in Table 5.

Table 5.

Glutamate spillover

To produce plateau potentials which have an all-or-none jump in the voltage, i.e. a threshold non-linearity (as suggested by existing in vitro experiments), we included glutamate spillover. A detailed study of glutamate spillover and its role in triggering plateau potentials is given in Trpevski et al. (2023), and here we briefly describe the model. Glutamate spillover is a phenomenon that occurs during repeated synaptic stimulation, when glutamate reuptake by the surrounding astrocytes is saturated and when the astrocytes may even reverse function from reuptake to glutamate releas (Malarkey and Parpura, 2008). When this happens in a synaptic cluster, glutamate spills over from the synaptic cleft into the extrasynaptic space, activating extrasynaptic NMDARs (eNMDARs). To model glutamate spillover, we place eNMDARs in the dendritic shaft, at the base of each spine in a cluster. Glutamate spillover is activated when a threshold level of clustered synaptic stimulation is reached – the “glutamate threshold”. In this study, the glutamate threshold is reached when 20 synapses with a weight w > 0.5 are activated (with 3 spikes each). Note that this threshold is equivalent to activating 10 synapses with a weight w > 1.0, or 40 synapses with a weight w > 0.25, i. e. the summed weight of all activated clustered synapses should be greater than 10.

When glutamate spills over, i) an activating spike to the eNMDARs is delivered 1.5 ms after the glutamate threshold is reached, ii) it stays much longer around the eNMDARs than in the synaptic cleft (T_dur is much longer), and iii) its concentration around the eNMDARs is much lower than in the synaptic cleft ([T]_max is much lower) (Table 5 and Trpevski et al. (2023)). Also, the eNMDARs represent about 64% of the total NMDA conductance in this study (Table 5).

Stimulation protocol

The stimulation protocol in Fig. 3A for presenting the patterns (in random order), which are followed by dopamine feedback, contains several additional details that we list here. First of all, when the distributed or clustered synapses are activated within the 35 ms stimulation window, 3 input spikes arrive to each synapse within this window (as explained in the caption of Fig. 3). This is because at physiological extracellular calcium concentrations, single spikes do not induce plasticity (at least in the spike-timing-dependent protocols in Inglebert et al. (2020)). Furthermore, with the backward rate constants from Dorman et al. (2018) for calcium unbinding from the buffers are much lower than experimentally measured values for calmodulin and calbindin (Faas et al., 2011; Nägerl et al., 2000). To avoid long simulation times on the computing cluster used for this study, 600 ms after a pattern is presented, we reset the values of the buffers, their bound forms to calcium, and the intra- and extracellular calcium concentrations back to their initial values before the first pattern is presented, thus “equilibrating” the system faster and shortening simulation time. As can be seen from Fig. 2C, by 600 ms the plateaus are over, and both [Ca]_NMDA and [Ca]_L-type are back to their baseline levels, so this abrupt reset of the system does not affect learning and synaptic plasticity. And as can be seen from Fig. 3C, it neither affects the baseline voltage in the soma and dendrites before the arrival of a future pattern.

Performance scores on the FBP and NFBP

In the FBP, a score of 50% means that the SPN is correct only half of the time. This happens if the SPN is silent for both the relevant and irrelevant pattern (correct output only for the irrelevant pattern), or if it spikes for both patterns (correct output only for the relevant pattern). A score of 100% indicates it always spikes for the relevant pattern, and is always silent for the irrelevant pattern, and the FBP is perfectly solved. The threshold score for solving the FBP is 75%, exemplified with the situation where the SPN is always silent for the irrelevant pattern, and spikes at least half of the time for the relevant pattern. A score of 0% would indicate that the SPN’s output is incorrect all the time, being silent for the relevant pattern, and spiking for the irrelevant pattern (this does not happen in the simulations).

In the NFBP, a score of 50% also means that the SPN output is correct only half of the time. An example of such a score is when the SPN is silent for all four patterns, or spikes for all four patterns. Similarly, a score of 100% indicates it always spikes for the relevant patterns, and is always silent for the irrelevant patterns. A score of 75% is exemplified with the situation where the SPN is silent for the two irrelevant patterns, and spikes for only one of the relevant patterns. The threshold score for solving the NFBP is 87.5%, where the SPN would also spike for the other relevant pattern at least half of the time.

Data availability

The current manuscript is a computational study, so no data have been generated for this manuscript. Modelling code is openly accessible at https://github.com/danieltrpevski/Plasticity/tree/metaplasticity

Acknowledgements

DT thanks Ana Kalajdjieva for illustrating the mouse brain. Simulations were performed on resources provided by the National Academic Infrastructure for Supercomputing in Sweden (NAISS) at PDC KTH.