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. However, 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 and CA3 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; Hunt et al., 2013; Kohda et al., 1995; Binda et al., 2016; Locatelli et al., 2021; Ji and Martin, 2012). 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 when followed by 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). (Note that to trigger LTD, activation of metabotropic glutamate receptors (mGluRs) is also required (Shen et al., 2008). This requirement assures that LTD is only elicited in active synapses where glutamate transmission occurs; Ca_v1.3 channels alone have an unspecific response – they can be active anywhere where the voltage is high enough.) A calcium threshold for triggering 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)). A calcium threshold for LTP 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). Most computational metaplasticity models have only one modifiable threshold (where synaptic activity below the threshold induces LTD, and above it induces LTP) (Yger and Gilson, 2015; Bienenstock et al., 1982; Jedlicka et al., 2015). On the other hand, the important class of cascade models, whose general form can be seen as having two thresholds, has shown that metaplasticity can solve (several versions of) the plasticity-stability dilemma – it protects stored information from being “deleted” (overwritten) by new synaptic activity, while also allowing for flexible relearning when this information is no longer useful (Fusi et al., 2005; Iigaya, 2016; Farashahi et al., 2017; Khorsand and Soltani, 2017; Jedlicka et al., 2022; Benna and Fusi, 2016). The cascade model is a stochastic model of a number of synapses (not placed on any neuron) that randomly experience plasticity events. In this article we study the role of metaplasticity in learning using models closer to biological neurons: a multicompartment model of a dSPN, and a simple, biologically-grounded learning rule describing plasticity of the dSPN cortico-striatal synapse, to which we add metaplasticity in the two thresholds (Fig. 1A). Using the linear and nonlinear feature binding problem (FBP and NFBP), whose structure, as will be explained later, is particularly suitable to emphasize the role of each threshold, we show that the two thresholds have specific, opposite and complementary roles in learning. Further, using reversal learning in the FBP and NFBP, we show that metaplasticity can also solve the plasticity-stability dilemma. Lastly, we show that while a single threshold is sufficient to solve the FBP, NFBP, and reversal learning, two calcium thresholds allow separate control over synaptic strengthening and weakening.

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). The binding problem arises because features are often processed in different brain regions, and possibly at different times, yet they need to be combined to represent/construct the perception of the object they correspond to. In its simplest form, the linear 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, top table). 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 NFBP consists of discriminating between four patterns: the neuron should spike for two patterns and remain 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 bottom 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 basal dopamine in the striatum low enough so that D₁Rs are not activated, enabling LTD (Yapo et al., 2017).

The reason why the FBP and the NFBP are particularly suitable to demonstrate the role of the calcium thresholds is that relevant and irrelevant patterns share features. In the FBP, the shared feature is the ‘strawberry’ shape (Fig. 1B, top). In the NFBP, both features of a pattern are shared, each feature being shared between one relevant and one irrelevant pattern (Fig. 1B, bottom). As a result, during learning, the synapses representing shared features will experience both dopamine signals (peaks and pauses), guiding them towards both strengthening and weakening. As will be shown, metaplasticity is crucial for “choosing” a direction, i.e. directing plasticity into either strengthening or weakening. In particular, we will see that metaplasticity in the LTD threshold is necessary for strengthening, while metaplasticity in the LTP threshold is necessary for weakening, i.e. that the threshold regulating one plasticity outcome is necessary for properly expressing the opposite plasticity outcome.

Lastly, as the input nucleus of the basal ganglia, the striatum integrates convergent cortical inputs from sensory, motor, limbic and associative areas, as well as thalamic and neuromodulatory inputs (Purves et al., 2018). Based on these inputs it initiates and invigorates goal-directed actions, such as reaching for one of the relevant patterns in the FBP and NFBP (Thura and Cisek, 2017). The cortico-striatal synapses are thought to be a major site for learning to initiate such actions, thus forming associations between environmental stimuli, internal states and rewarding outcomes (Filippo et al., 2009). Because of this, despite being suitable tasks for studying metaplasticity, the FBP, NFBP, and reversal learning are also relevant tasks for the striatum’s function. Throughout this article, feature binding is illustrated with a commonly used example from the visual system, although diverse features from the other brain areas would be involved in the striatum (an example is given in Figure 1–figure supplement 1).

Before continuing to the results, we note that we recently participated in a related study with a different goal – whether SPNs can solve the NFBP (Khodadadi et al., 2025). That study used the same neuron model in a regime with much less supralinear integration, and with a different, biologically more detailed learning rule, which incorporated metaplasticity in the LTP threshold only (operating according to a different mechanism and where the evoking calcium signal does not diffuse across neuronal compartments). On the other hand, the goal of this article is to study the role of metaplasticity in the two calcium thresholds in learning, for which we use a simpler, yet biologically-based learning rule where both thresholds are plastic.

Results

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

We first demonstrate linear and supralinear integration in the dSPN model, which are the substrates for solving the FBP and NFBP. Figure 2 compares linear integration of synaptic inputs distributed across the dendrites with supralinear 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 (calcium concentrations arising from NMDARs and L-type calcium channels, respectively) 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 (due to the distance between them, spines cannot influence each other’s voltage, 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 experimental studies have suggested that glutamate spillover, which activates extrasynaptic NMDARs, promotes plateaus or is even necessary to evoke them (Suzuki et al., 2008; Chalifoux and Carter, 2011; Oikonomou et al., 2012). 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. Figures 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 because they trigger a different plasticity outcome ([Ca]_NMDA is needed for LTP and [Ca]_L-type is needed for LTD). Also, [Ca]_NMDA diffuses in the neuron, while [Ca]_L-type does not, since it is assumed to be restricted to microdomains, a form of highly localized signaling often mediated by L-type calcium channels in neurons (Berridge, 2006; Parekh, 2008; Chen and Sabatini, 2012). 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 up to 20 synapses). Once glutamate spillover occurs (here at a cluster size above 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 nonlinearity being caused by the plateau (Figs. 2D₁-D₂, red lines). The amplitudes of [Ca]_NMDA and [Ca]_L-type before spillover reflect the voltage dependencies of NMDARs and L-type channels (Figs. 2D₃-D₄, red lines for up to 20 synapses), and the largest nonlinearity is again caused by the plateau. The jump in all four quantities in Fig. 2D is an indication of a “threshold supralinearity”. When a critical 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. This type of supralinear behavior should in principle be sufficient to solve the NFBP (Tran-Van-Minh et al., 2015).

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. (In the somatic voltage there is also 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

Before demonstrating the role of metaplasticity using the FBP and NFBP, a learning rule that can solve these two tasks is needed. 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. ([Ca]_NMDA and [Ca]_L-type are tracked separately in the SPN model, since they produce a different plasticity outcome. Also, as mentioned above, [Ca]_NMDA diffuses in the neuron, while [Ca]_L-type is assumed to be restricted to microdomains, a form of highly localized signaling often mediated by L-type calcium channels in neurons (Berridge, 2006; Parekh, 2008; Chen and Sabatini, 2012).) Synapses can be increased to a maximal level of w_max, and decreased to a minimal level of w_min, and weight updates are 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. (Note that Δt is neither the timing between pre- and postsynaptic spikes as in spike-timing-dependent rules – this rule does not depend on somatic spiking, nor is it the time until dopamine feedback is provided. It is simply the time step for numerical integration.) 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). Similarly, in hippocampal CA1 neurons, local dendritic activity is sufficient for synaptic plasticity (Bittner et al., 2017).

Metaplasticity is also implemented in a simple way, given with Eq. 2. In most of the article, 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 formulation of metaplasticity is very similar to the sliding threshold in the well-studied BCM learning rule, where the threshold follows a quantity derived from synaptic activity (Bienenstock et al., 1982). 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 under the conditions in Eq. 1.

Such “thresholded” metaplasticity, which requires the same conditions to occur as synaptic plasticity, cannot solve reversal learning, a task used to demonstrate the plasticity–stability dilemma. Because of this, when simulating reversal learning, we relaxed the conditions for metaplasticity so that it can be triggered by any calcium level in activated synapses (i.e. no need for calcium to be above a threshold as 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 intervals until reward delivery and the next pattern were chosen to shorten simulations; behavioral timescales would likely be longer; see the Methods section for more details.) The learning rule is always active (always “on”), avoiding separate training and testing phases sometimes used in other plasticity rules.

Figure 3.

Before studying the effects of each calcium threshold on learning, we first show that the synaptic plasticity and metaplasticity rule can solve the FBP and NFBP (Fig. 3). 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 or inhibition (Legenstein and Maass, 2011; Trpevski et al., 2026).

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 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 four patterns in the task, i.e. twice as many as in the FBP.) When plateaus are evoked by the relevant patterns, they drive somatic spiking (as indicated above in Figs. 3C₂, C₃). In contrast, an irrelevant pattern stimulates only half of the strengthened synapses on one dendrite, which are the synapses representing the shared feature with the relevant pattern (the ‘strawberry’ feature in dendrite 1 in Figs. 4B₂, B₃ and the ‘yellow’ feature in dendrite 2 in Fig. 4B₄). 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 shared strengthened synapses are protected from weakening as a result of metaplasticity. Finally, we also show that the NFBP cannot be solved by linear integration – with distributed synapses only one relevant pattern can be stored by the neuron (Figure 4–figure supplement 1).

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 2 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 2 and also highlighted with the light blue regions in Figure 4–figure supplement 2B₃). 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). Note that the LTP threshold θ_LTP can also go above the upper plasticity threshold Θ_LTP (θ_LTP and Θ_LTP are independent), reflecting the assumption that different molecular circuitry implements these two thresholds. (For example, experiments in hippocampal CA1 pyramidal neurons have demonstrated that the two different voltage thresholds for LTD and LTP can reverse their relative positions as a result of metaplasticity: the voltage threshold for LTD can rise to -20 mV, going above the voltage threshold for LTP, which reduced to -30 mV (Ngezahayo et al., 2000)).

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. 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 2, and it is also highlighted with the green and brown regions in Figure 4–figure supplement 2B₃, 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 NFBP the weakened synapses and their thresholds can be updated during both LTP and LTD, because these synapses are also activated by a relevant pattern (i.e. are shared with 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, plasticity and metaplasticity cannot be triggered, and the thresholds, as well as the synapses, are not updated anymore. In other words, once the thresholds reach the evoked calcium amplitudes, synapses are stabilized. In effect, metaplasticity acts as a lock on the synapses: synapses are flexible for learning in the time it takes for the thresholds to reach the calcium amplitudes, and are stabilized once that occurs. Accordingly, we have assumed that at the beginning of learning thresholds start at a low value, so that synapses are initially flexible to store information. All of this will be much more evident in the next section, where we study the role of metaplasticity.

We stress once again that in the FBP with supralinear integration, one of the strengthened synaptic clusters belongs to the shared feature (the ‘strawberry’ cluster in Figs. 4A₂, B₂), and in the NFBP both strengthened synaptic clusters belong to shared features (the ‘red’ and ‘strawberry’ clusters in dendrite 1 in Figs. 4A₃, B₃, and the the ‘yellow’ and ‘banana’ clusters in dendrite 2 in Figs. 4A₃, B₄). Importantly, the LTD threshold for these synapses is raised because of the plateaus (Figs. 4D₂–D₄). 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 seconds are far below their ‘strawberry’ threshold, and dark red dots are far below their ‘red’ threshold in Fig. 4D₃; similarly yellow and dark yellow dots in Fig. 4D₄ located just above the ‘strawberry’ threshold after 100 seconds are far below their ‘yellow’ and ‘banana’ thresholds, respectively; dots are also highlighted in the pink region of Figure 4–figure supplement 2B₃). In this way, these strengthened synapses belonging to a shared feature 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.

The effects of parameters in the learning rule

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 from saturating at their maximal levels. Without it, the FBP is still solved, even though weights attain their maximal values (Figure 4–figure supplements 3B₁ and 3E₁). 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 3A₁). 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 3A₂, 3B₂, 3B₃ and 3E₂). The maximally strengthened synapses also drive more weakening in the weakened synapses, compared to when an upper threshold is present (Figure 4–figure supplement 3B, cf. weakened weights in Fig. 4B₂– B₄). This is due to the elevated [Ca]_L-type evoked when an irrelevant pattern activates the maximally strengthened synapses (shared “strawberry” synapses in Figure 4–figure supplements 3B₁, B₂ and shared “yellow” synapses in Figure 4–figure supplement 3B₃). 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 3D₁, D₂ and “strawberry” synapse in Figure 4–figure supplement 3D₃, 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 3C when compared to Figs. 4C₂–C₄).

The upper threshold Θ_LTP is necessary to solve the FBP and NFBP because there is no single value of the maximal weight w_max with which both tasks are solved simultaneously, as can be seen from the parameter scan for w_max in Figure 4–figure supplement 4 done in the absence of Θ_LTP. The FBP with supralinear integration and the NFBP are first solved for (due to the w_max threshold for glutamate spillover being between and in this study, and spillover evoking plateaus), but this value is not high enough to solve the FBP with linear integration (Figure 4–figure supplement 4, second column). A value of is needed to solve the FBP with linear integration, which, on the other hand, is too high to solve the NFBP (the performance on the NFBP drops to the threshold score for solving it in Figure 4–figure supplement 4, fourth column). All other values for increase the performance on the FBP with linear integration, and cause faster learning of the FBP with supralinear integration, but lower the performance on the NFBP due to spiking for the irrelevant patterns, as shown previously in Figure 4–figure supplement 3A₂. This suggests that if a range of w_max values exists that solves all three tasks, it is very narrow, and such a solution is not robust to variations in w_max.

The maximal weight w_max prevents unbounded weight growth, ensuring stability of the neuron’s firing rate in the FBP with linear integration (Figure 4–figure supplement 5). Without w_max, the weights grow until [Ca]_NMDA saturates in the spines (which does not occur for most synapses within the simulation time in Figure 4–figure supplement 5B; also the weights grow to unphysiological values, which of course does not happen in reality). Thus, in the FBP with linear integration, the neuron enters depolarization block when stimulated by a pattern after learning (Figure 4–figure supplement 5A₁). On the other hand, in the FBP with supralinear integration, firing rate stability is ensured by the plateau potentials, since a plateau’s voltage is limited by the NMDAR reversal potential, regardless of how high the synaptic weights are (Figure 4–figure supplement 5A₂). This is part of the plateaus’ function to provide dynamic range compression, as shown experimentally in Figs. 3E, 4D and 12 in Oikonomou et al. (2012).

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 6). However, if the learning rate η is very low or very high, the NFBP cannot be solved: a very low η causes spiking for the irrelevant patterns because synapses are not weakened enough (Figure 4–figure supplement 7), and a very high η causes large updates to the synaptic weights, disrupting learning due to the similarly large changes in calcium, which can easily fall below the thresholds (Figure 4–figure supplement 8). Within a certain range, 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 9). If the metaplasticity rate is too low, it takes a long time for the thresholds to reach the calcium amplitudes, and as a result, synapses take a longer time to stabilize in the FBP with linear integration and in the NFBP (Figure 4–figure supplement 10B₁, 10B₃, 10B₄). Synapses stabilize quickly in the FBP with supralinear integration because it is sufficient that one of the clusters is strengthened enough to trigger plateaus (Figure 4–figure supplement 10B₂). Learning is slowed down in the NFBP – more patterns are needed before the same performance level is reached (Figure 4–figure supplement 10E₃, cf. Fig. 3D₃), while learning in the FBP is unaffected (Figure 4–figure supplements 10E₁, E₂, cf. Figs. 3D₁, D₂). On the other hand, if the metaplasticity rate is too high, the thresholds reach the respective calcium levels quickly, shortening the period where synapses are flexible and preventing them from being changed very much, i.e. they are quickly stabilized (Figure 4–figure supplement 11B–D). This results in more variable performance on the FBP with linear integration because weights are not strengthened enough (Figure 4–figure supplement 11E₁), and in reduced performance on the NFBP because weakened clusters are not weakened enough, causing spiking for the irrelevant patterns (Figure 4–figure supplement 11E₃).

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. As mentioned in the introduction, the FBP and NFBP are particularly useful for demonstrating the role of each threshold because the patterns in the tasks share features. In the FBP, the relevant and irrelevant pattern (the red and yellow strawberry) share the shape feature (the feature “strawberry”). This means that during learning, the synapses representing this feature experience both strengthening and weakening (e.g. visible at the beginning of learning in Figs. 4B₁, B₂), and that strengthening is somehow “chosen”. Fixing the LTD threshold (to the initial low value in the simulation) shows that the shared synapses cannot strengthen properly, i.e. cannot stabilize (Figs. 5B₁, B₂). This happens because there is no metaplasticity in the LTD threshold – it is fixed, i.e. it never rises to follow [Ca]_L-type. As a result, whenever an irrelevant pattern arrives, it always causes higher [Ca]_L-type than the fixed LTD threshold, and can always weaken the shared synapses, despite them also being strengthened by the relevant pattern (Figs. 5D₁, D₂ – LTD thresholds stay fixed during learning, and [Ca]_L-type is always above θ_LTD). The other two features are not shared – the feature “red” is only activated by the relevant pattern and, the feature “yellow” only by the irrelevant pattern. As a result, they can only ever experience plasticity in one direction: the feature “red” is strengthened, and the feature “yellow” is weakened. The fixed LTD threshold has no influence on the strengthened feature (“red”). However, in the FBP with supralinear integration, where the cooperative voltage effects of the clustered synapses produce higher calcium elevations, it does influence the weakened feature (“yellow”): because the LTD threshold is fixed to a low value, the evoked [Ca]_L-type is high enough to decrease the weakened synapses much more compared to when metaplasticity in the LTD threshold is active (Figs. 5B₁, B₂, cf. “yellow” synapses in Figs. 4B₁, B₂).

Figure 5.

Regarding the performance, in the FBP with linear integration, despite the shared synapses not stabilizing, the strengthening is enough to drive somatic spiking (Fig. 5A₁), and the task is solved but with a lower performance (Fig. 5E₁). On the other hand, in the FBP with supralinear integration, because the shared “strawberry” synapses do not stabilize, the clustered synapses are not strong enough to evoke plateaus (Fig. 5A₂), and the FBP is not learned (Fig. 5E₂).

The effect of the fixed LTD threshold on the NFBP is analogous, but even more prominent because in the NFBP, a relevant pattern shares its features with both irrelevant patterns (e.g. “red strawberry” shares the color feature with “red banana” and the shape feature with “yellow strawberry”). This means that during learning, both features of a relevant pattern experience strengthening and weakening. Hence, when the LTD threshold is fixed, none of their synapses stabilize, and the relevant patterns are not stored in the dendrites (neither “red” and “strawberry” in dendrite 1, Fig. 5B₃, nor “yellow” and “banana” in dendrite 2, Fig. 5B₄). As a result, they cannot evoke plateaus, and the NFBP is not learned (Figs. 5A₃, E₃). The LTP threshold follows the level of [Ca]_NMDA and also does not stabilize (Figs. 5C₃, C₄). As with the FBP, this happens because there is no metaplasticity in the LTD threshold, and irrelevant patterns always cause higher [Ca]_L-type than the fixed LTD threshold (Figs. 5D₃, D₄). This means that despite being strengthened, these synapses 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 prior 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–D₄ (e.g. light red dots in pink region in Figure 4–figure supplement 2B₃ are below the elevated “strawberry” threshold).

Without metaplasticity in the LTD threshold, strengthened synapses are not protected from weakening. 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.

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. This effect can be seen in a shared feature which ultimately needs to be weakened, but which experiences a drive to both strengthen and weaken during learning. There is no such feature in the FBP (the only shared feature, “strawberry”, needs to be strengthened), so fixing the LTP threshold has no effect on solving this task (Figs. 6A₁–E₁ and 6A₂–E₂, cf. Figs. 3A_1,2–C_1,2, 4A_1,2–D_1,2). Nevertheless, there are such features in the NFBP, which is why it is a suitable task to demonstrate the role of the LTP threshold. In fact, each irrelevant pattern in the NFBP also shares its two features with the relevant patterns, and, as a result, they also experience both strengthening and weakening during learning (e.g. the features “yellow” in dendrite 1 and and “strawberry” in dendrite 2 in Fig. 4A₃).

Figure 6.

If the LTP threshold is fixed, the features belonging to the irrelevant pattern in each dendrite cannot be weakened properly (“yellow” synapses in dendrite 1 in Fig. 6B₃ and “strawberry” synapses in dendrite 2 in Fig. 6B₄). 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. 6C₃, C₄), triggering synaptic strengthening even if synapses have previously been 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 Fig. 4C₃ and for “strawberry” in Fig. 4C₄), 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 Fig. 4B₃ and for “strawberry” in Fig. 4B₄ are successfully weakened, while in Figs. 6B₃, B₄ they are not. (The LTD threshold behaves as usual for the strengthened synapses, protecting them from weakening, as shown in Figs. 6B₃, B₄ and 6D₃, D₄. However, for the weakened synapses, it does not stabilize.) Without metaplasticity in the LTP threshold, the NFBP is on average not solved (Fig. 6E₃). Despite both relevant patterns being stored by each dendrite, the performance is lowered because the SPN sometimes also spikes for the irrelevant patterns (Fig. 6A₃). Also, in the input configurations containing four features in one dendrite, the average performance even drops below 50 % (green trace in Fig. 6E₃). In summary, by preventing synapses from strengthening, the threshold for strengthening allows proper synaptic weakening.

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. This time we start with the NFBP, since that is the task where the role of both thresholds can be demonstrated. 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 strengthened, 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 the “yellow” 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 by 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 metaplasticity would work. As shown in Figs. 6A₁–E₁ and 6A₂–E₂, even when the LTP threshold is fixed, it does not have an effect on solving the FBP (because there are no shared features in the FBP which ultimately need to be weakened). 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 metaplasticity in the LTD threshold is not enough to protect the shared 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 shared “strawberry” synapses are not strengthened properly (even though, for example, their weights stabilize when using supralinear integration, Figure 7–figure supplement 1B). Similarly to Figs. 5A₁–E₁ and 5A₂–E₂, 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 with linear integration are similar to those of having partial metaplasticity (Figure 8–figure supplement 1). Again, the synapses encoding the shared feature cannot stabilize (synapses for “strawberry”, Figure 8–figure supplement 1B), only moderately affecting the performance (Figure 8–figure supplement 1A₁ and 1E₁). No metaplasticity in the FBP with supralinear integration also causes synapses encoding the shared to not stabilize (Figure 8–figure supplement 1A₂). However, compared to partial metaplasticity, the unstable shared synapses sometimes evoke plateaus as they oscillate, on average solving the task (Figure 8–figure supplement 1E₂). In summary, this shows that in the FBP, metaplasticity is important for robust, consistently high performance on the task.

Reversal learning can be solved when relaxing the conditions for metaplasticity or introducing a signal that resets the thresholds

We have so far demonstrated a situation where the calcium thresholds are initially set to a low value, making the synapses “flexible” for learning in the time it takes for the thresholds to reach the [Ca]_NMDA and [Ca]_L-type amplitudes. Once the thresholds have elevated and reach the calcium amplitudes, the synapses are effectively “locked”, providing stable information storage. Conversely, fixing the calcium thresholds to study metaplasticity (in Figs. 5 and 6) made the synapses permanently flexible for learning and no information could be stored. This illustrates the role of metaplasticity as a lock on the plasticity process, and that it plays a role in solving the plasticity–stability dilemma, which means that synapses should “both be stable enough to store information and be plastic enough to accommodate new learning” (Jedlicka et al., 2022). Reversal learning is a straightforward example of the plasticity–stability dilemma applied to the FBP and NFBP: after the tasks have been learned, we switch the reward policy so that relevant patterns become irrelevant (and are followed by dopamine pauses), and vice versa, irrelevant patterns become relevant (and are followed by dopamine peaks). To solve this task in the (N)FBP, the neuron should keep the learned changes in (one of the) the shared strengthened feature(s) (e.g. the shared “strawberry” is necessary for both the “red strawberry” and “yellow strawberry” patterns), and change the weights of the other features (e.g. “red” should be weakened and “yellow” should be strenghtened to reverse the neuron’s response from “red strawberry” to “yellow strawberry”).

We first show that “thresholded” metaplasticity cannot solve reversal learning (Fig. 9). The reward policy is reversed in the middle of the simulation, and all three tasks are solved before the reversal (blue voltage traces in Fig. 9A, and score in Fig. 9E before reversal). After reversal, because metaplasticity is only activated if a calcium signal is above its threshold (and followed by dopamine feedback), this means that the strengthened features cannot be weakened properly once their calcium levels fall below their LTD thresholds (strengthened synapses in Figs. 9B₁–B₄ weaken only slightly because their [Ca]_L-type levels fall below the LTD thresholds after weakening, shown in Figs. 9D₁–D₄; this fall in [Ca]_L-type is very prominent with supralinear integration because after weakening, no plateaus can be evoked). Neither can the weakened features be strengthened in most of the cases, because the previosuly irrelevant patterns, which are now rewarded and relevant, rarely cause [Ca]_NMDA above the LTP thresholds (calcium dots in Figs. 9C₁–C₄ largely stay below the LTP threshold of the weakened synapses). The FBP and NFBP are successfully learned before the point of reward policy reversal, while after reversal, the synapses cannot adapt to the new situation. As a result, in the end there is no spiking for the initially irrelevant patterns, nor is there spiking for the initially relevant patterns that were learned before reversal (Figs. 9A₁–A₃, black traces, and Figs. 9E₁–E₃). As expected, “thresholded” metaplasticity permanently locks the patterns in the dendrites, making the rule inflexible to learn new pattern-outcome associations if the reward policy changes.

Figure 9.

To solve reversal learning, we demonstrate two different approaches. One is to allow the thresholds to be lowered again, i.e. prevent them from stabilizing at an elevated level forever. To do this, we simply relax the conditions for metaplasticity so that any calcium levels in the active synapses can cause metaplasticity, i.e. calcium no longer needs to be above a threshold level for metaplasticity to occur (note that those conditions still hold for synaptic plasticity, as in Eq. 1). This approach has been taken in an extension of the cascade model, which in its original form cannot solve reversal learning. (In the generalized model the threshold is allowed to move in both directions (Farashahi et al., 2017).) We first show that such “relaxed” metaplasticity can solve the FBP and NFBP (Figure 9–figure supplement 1). In this case, the thresholds always follow any calcium levels evoked when a synapse is activated by presynaptic input, allowing them to lower in value. When the variability in the calcium amplitudes is small, the fluctuations in the thresholds are also small, as occurs in the FBP with linear integration (Figure 9–figure supplement 1C₁, D₁), the strengthened “red” synaptic cluster in the FBP with supralinear integration (Figure 9–figure supplement 1C₂, D₂, dark red traces) and the weakened synaptic clusters in supralinear integration (Figure 9–figure supplement 1C_{3, 4}, D_3,4). Conversely, when the variability in the calcium amplitudes is large, as occurs in the shared synapses that need to be strengthened when using supralinear integration, the thresholds show large fluctuations (Figure 9–figure supplement 1B_2–4–D_2–4). Strictly speaking, with “relaxed” metaplasticity the thresholds never stabilize, which is particularly evident for the shared strengthened clustered synapses with large variability in the calcium amplitudes (Figure 9–figure supplement 1C_2–4, D_2–4). Despite the fluctuations, the thresholds perform their function equally well: the LTD thresholds are elevated enough to protect the strengthened synapses from weakening, and the LTP thresholds follow [Ca]_NMDA in the the weakened synapses to prevent them from strengthening. Note that because of the relaxed metaplasticity conditions, the LTD thresholds for the weakened synapses are not elevated as with “thresholded” metaplasticity (Figure 9–figure supplement 1D₂–D₄, cf. Fig. 5D₂–D₄). Hence, the weakened synapses decrease almost to the minimal level w_min (Figure 9–figure supplement 1B₂–B₄). Finally, “relaxed” metaplasticity allows to overcome initially high calcium thresholds set before learning (Figure 9–figure supplement 2). With “thresholded” metaplasticity, if the thresholds are initialized to values higher than the calcium signals, the synapses will remain locked and no learning will occur (Figure 9–figure supplement 2B₁, B₂). “Relaxed” metaplasticity can instead adapt the thresholds, thus unlocking the synapses and allowing learning (Figure 9–figure supplement 2B₃, B₄).

Relaxing the conditions for metaplasticity allows the rule to solve reversal learning in both the FBP and the NFBP (Fig. 10). The key to solving it with this approach is that calcium thresholds which have been raised while learning the FBP and NFBP are free to be lowered again after reversing the reward policy. In the FBP, after reversal, the LTD thresholds for the features “red” and “strawberry” are lowered, so these two features are weakened (Figs. 10B₁, B₂ and 10D₁, D₂, these weights and thresholds are weakened immediately after reversal). Simultaneously, “yellow strawberry” evokes higher [Ca]_NMDA than the “yellow” LTP threshold because in the initial stages of reversal the feature “strawberry” is still strong, despite being weakened (Figs. 10C₁, C₂, LTP threshold for “yellow” rises to follow the evoked [Ca]_NMDA). When the feature “yellow” is strengthened enough, it elicits strong [Ca]_NMDA together with the “strawberry” feature, causing the latter to strengthen again (Figs. 10B₁, B₂, “yellow” synapses increase after reversal, and “strawberry” synapses follow after some time). The LTD thresholds for “yellow” and “strawberry” become elevated and protect the two strengthened features from weakening (Figs. 10D₁–D₃). Conversely, the LTP thresholds for “red” follow [Ca]_NMDA and prevent this feature from strengthening. In this way, in the FBP, the initially relevant pattern “red strawberry” stops evoking spiking (through linear integration at the soma or via a plateau), and the neuron starts responding to the initially irrelevant pattern “yellow strawberry”, thus solving the task (blue voltage traces before reversal learning and black traces after reversal learning in Fig. 10A₁, A₂, and performance scores in Fig. 10E₁, E₂).

Figure 10.

The same occurs in the NFBP – the neuron stops responding to the two initially relevant patterns (Fig. 10A₃, blue voltage traces before reversal show a plateau potential in each dendrite evoked by a relevant pattern, and black traces after reversal show no plateaus and no somatic spiking for these two patterns). However, since there is no structural plasticity, the available synaptic connectivity in this example (Fig. 3A₃) allows for only one of the initially relevant patterns to be stored (in both dendrites, as seen from the synaptic weights in Figs. 10B₃, B₄; also, black voltage traces after reversal show plateaus in both dendrites for “yellow strawberry” in Fig. 10A₃). This is also reflected in the performance, which shows that before reversing the reward policy, the NFBP is solved, while after the reversal the neuron responds to only one pattern (Figs. 10E₃). To show that both initially irrelevant patterns can be learned in reversal learning, we also simulated the task on input configurations that are innervated by only one initially relevant pattern, but both irrelevant patterns (Figure 10–figure supplements 1 and 2). An example of such a configuration is given in Figure 10– figure supplement 2A, where only one initially relevant pattern (“red strawberry”) can be stored in both dendrites, but both initially irrelevant patterns (“yellow strawberry” and “red banana”) innervate dendrites 1 and 2, respectively. During learning, the neuron first learns to respond to “red strawberry”, and after reward policy reversal spikes to both “yellow strawberry” and “red banana”, elicited via plateau potentials in dendrites 1 and 2, respectively (Figure 10–figure supplement 2B, blue traces before reversal, and black traces after reversal, and Figure 10–figure supplement 2E, score of 75% before reversal, and close to 100% after reversal). Finally, there are a total of four input configurations in group 1 in Figure 3–figure supplement 1 and Figure 10–figure supplement 2 (marked with a star) which can store both relevant patterns and both irrelevant patterns. Reversal learning using (one of) these is shown in Figure 10–figure supplement 3, demonstrating that given the appropriate structural connectivity, reversal learning from both relevant to both irrelevant patterns can be done.

Another approach to solve reversal learning is to assume that another part of the brain detects the change in the reward policy and sends a signal to temporarily “reset” the permanently stabilized thresholds to a low value, thus “unlocking” the synapses, i.e. making the weights flexible again to relearn the new pattern-reward associations (Fig. 11). (This approach has also been taken in another extension of the original cascade model (Iigaya, 2016).) The noradrenergic system, particularly the locus coeruleus, has been implicated in detecting unexpected changes in environmental conditions and facilitating behavioral adaptation, including during reversal learning (Jordan, 2024; Sara, 2009; Pajkossy et al., 2023). The striatum receives only limited projections from the locus coeruleus, but SPNs express adrenergic receptors which modulate phosphorylation of DARPP-32, an important protein regulating cortico-striatal synaptic plasticity (Hara et al., 2010), and hence could be affected by such signals about environmental changes.

Figure 11.

Two calcium thresholds provide separate control over LTP and LTD

In this section we ask the question what is the use of two separate calcium thresholds for LTP and LTD? The FBP, NFBP and reversal learning can be solved just as well with a single plasticity threshold (Fig. 12 and Figure 12–figure supplements 1 and 2). To show this we simulated a scenario where calcium from both NMDA and L-type channels is tracked with a single variable, [Ca] (rather than two separate ones), and diffuses axially across the spines and dendrites, and the same [Ca] signal is responsible for LTP and LTD when it crosses a single calcium threshold (Eqs. 4 and 5). This is not biologically realistic in dSPNs, but allows us to compare the functionality of two thresholds versus just one. As before, the dopamine signal determines the direction of synaptic plasticity:

Figure 12.

and we use both “thresholded” metaplasticity (θ is updated only when [Ca] goes above it, Fig. 12) and the relaxed condition for metaplasticity (θ is updated for any [Ca] level in active synapses, Figure 12–figure supplement 1). In both cases, the dynamics of the weights and thresholds are analogous to those when two separate calcium thresholds are used. As before, no metaplasticity (fixing the threshold to a low value) results in the synapses representing shared features not stabilizing (Figure 12–figure supplement 3). With one calcium threshold, in the case of the NFBP none of the synapses stabilize (as occurs when both thresholds are fixed in Fig. 8). Comparing Figure 12–figure supplement 3B₃, B₄ to Figs. 5B₃, B₄ and 6B₃, B₄ where just one of the two calcium thresholds is fixed, we see that having two separate thresholds allows for separate control over synaptic strengthening and weakening.

The reason why a single plasticity threshold works as well as two separate ones on the FBP, NFBP and reversal learning is that [Ca]_NMDA and [Ca]_L-type are driven by the same variable synaptic voltage. Both their amplitudes, which are used to compare against the thresholds and the full [Ca]_NMDA and [Ca]_L-type time-courses are highly correlated (Figure 12 – figure supplement 4), meaning that the two signals carry the same information, and it is not necessary to have two separate calcium variables to track that information. In this study, we have triggered metaplasticity in both thresholds using the same signal (a calcium source should be above its threshold, followed by a dopamine signal; or simply a dopamine signal with “relaxed” metaplasticity). If metaplasticity in the LTP and LTD thresholds is triggered under different conditions for each, then two separate thresholds could be useful.

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 used the FBP and NFBP because the patterns in these tasks share their features, and the role of each threshold is clearly demonstrated on the synapses representing these shared features. Metaplasticity slides the thresholds towards the calcium amplitudes evoked by arriving patterns. Two different metaplasticity rules were tested: i) one where calcium levels need to be above the thresholds to trigger metaplasticity (“thresholded” metaplasticity), and ii) one where any calcium levels trigger metaplasticity (relaxed conditions). The results showed that:

1. The FBP and NFBP can be solved with this simple plasticity and metaplasticity rule, which is required in order to demonstrate the role of each calcium threshold. With this rule, metaplasticity is necessary for solving the NFBP, whereas in the FBP it ensures robust and consistently high task performance.

2. Metaplasticity enables synapses undergoing both LTP and LTD during learning to selectively express (converge to) only one form of plasticity (either LTP or LTD).

3. 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.

4. 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.

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

6. Relaxing the conditions for metaplasticity so that it is triggered by any calcium levels allows reversal learning in the FBP and NFBP to be solved, demonstrating metaplasticity’s importance for solving the plasticity–stability dilemma. Alternatively, assuming a signal exists that can reset the calcium thresholds, reversal learning can be solved without relaxing the conditions for metaplasticity.

7. Using a single calcium threshold instead of two separate ones also solves the FBP, NFBP and reversal learning, but does not allow for separate control over synaptic strengthening and weakening.

The role of metaplastcity in this and other studies

In the original BCM rule, metaplasticity prevents runaway growth or collapse of the synaptic weights, which is one of its most commonly cited roles (Bienenstock et al., 1982; Abraham, 2008; Hulme et al., 2013). This is achieved by using a nonlinear function in the update rule for the threshold. Conversely, here we used a linear function in the metaplasticity rule for sliding the thresholds towards the calcium amplitudes, so metaplasticity by itself cannot prevent the weights from unbounded growth (unless that growth produces saturation in the evoked calcium amplitude). Instead, metaplasticity has a different role in this study – it enables synapses exposed to competing LTP and LTD drives to converge to one stable plasticity state, i.e. either LTP or LTD – without metaplasticity, the synapses never stabilize. Although not explicitly demonstrated, the same role of metaplasticity can be inferred from the dynamics of the cascade model (Fusi et al., 2005; Farashahi et al., 2017). In fact, the calcium thresholds in the current rule are a possible mechanistic implementation of the abstract hidden metaplastic states in the cascade model: threshold sliding is analogous to moving up and down the cascade of states, making synaptic plasticity easier or more difficult to occur. (Hence the similar role of metaplasticity in both studies.)

More specifically, this study shows that 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). In addition to separately controlling the two plasticity processes, two thresholds allow implementing a different update rule for each threshold (a case which we have not studied here). Generally speaking, two thresholds would be useful if the different calcium concentrations that they track are uncorrelated, i.e. exhibit separate dynamics, which does not seem to be the case for the NMDA and L-type channels in the dSPNs. They could also be useful if metaplasticity in each threshold is triggered by a different signal (a case which we also did not study here, but only touched upon with partial metaplasticity). Having two thresholds might also contribute to increased robustness of the machinery for synaptic plasticity across different conditions, a common feature of biological systems (Kitano, 2004; Wollman, 2018).

Finally, in a range of studies using the same threshold for all synapses, derived from the neuron’s activity (similarly to Bienenstock et al. (1982)), metaplasticity provides a homeostatic, heterosynaptic effect which helps to keep a neuron’s firing rate stable (Jedlicka et al., 2015; Zenke et al., 2013). The learning rule in this article does not incorporate such effects. As a result, without an upper bound on the synaptic weight, synapses would increase indefinitely in the case of FBP with linear integration, causing a depolarization block in the neuron. Similarly, without the upper threshold for LTP, they would also increase indefinitely when using supralinear integration. However, in the latter case the plateau potential provides stability of the neuronal firing rate, since the dendritic voltage is limited by the NMDAR reversal potential.

Directions for future studies

This study also provides a basis for exploring other metaplasticity rules. Here 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.

Some examples of different metaplasticity mechanisms are known from other brain areas. In the Schaffer collateral synapses to the CA1 hippocampal 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). Additionally, weakening these synapses lowers the LTP threshold and raises the LTD threshold, while weakening does the opposite, i.e. lowers the LTD threshold and likely raises the LTP threshold (Ngezahayo et al., 2000). 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). Adapting the learning rule to incorporate the metaplasticity mechanisms in different neurons will allow to study whether metaplasticity supports other specific functions in various brain regions.

In addition to signals arising from the local neuronal microcircuitry, it is interesting to see if global signals reaching many brain areas might also have a role in triggering metaplasticity. The noradrenergic neuromodulatory system is involved in detecting unexpected changes in learned pattern-outcome associations, and has been proposed to transiently interrupt or reset ongoing neural processing (Jordan, 2024; Sara, 2009; Pajkossy et al., 2023; Yu and Dayan, 2005; Dayan and Yu, 2006). Noradrenaline is known to modulate synaptic plasticity through adrenergic receptors, such as the phosphorylation level of DARPP-32 (Hara et al., 2010), an important protein for corticostriatal synaptic plasticity. Hence, it is interesting to ask whether noradrenergic signaling can act as a global metaplastic signal that transiently modifies plasticity thresholds.

An important direction for future work is to develop a learning rule that also models structural plasticity. Since NMDAR-dependent supralinear integration relies on synaptic clustering within a short dendritic segment, or more widely, within a whole branch (Losonczy and Magee, 2006; Branco et al., 2010), a learning rule would ideally implement mechanisms that can give rise to such clusters. Synaptic clustering occurs already in development, and learning in the motor cortex, for example, builds on existing functional clusters by growing new spines into them (Winnubst and Lohmann, 2012; Hedrick et al., 2022). Motor learning also strengthens the synaptic connections from the motor cortex to the striatum, which are themselves clustered (Hwang et al., 2022). A rule incorporating structural plasticity could then also include thresholds specific to structural plasticity (Ueda et al., 2022).

Assumptions in the learning rule

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). Finally the astrocytically secreted S100B protein could be involved in limiting the amount of LTP within certain bounds (Jones, 2015).

Insights into nonlinear computation by neurons

Finally, this study gives insights into nonlinear computation by single neurons, which have a variety of mechanisms for nonlinear input processing, giving them computational abilities similar to those of multilayer artificial neural networks (Gidon et al., 2020; Poirazi et al., 2003; Jones and Kording, 2021; Major et al., 2013; Antic et al., 2010; Tran-Van-Minh et al., 2015). Metaplasticity allows dSPNs to perform the NFBP, a computationally difficult task that relies on supralinear integration by the clustered synapses. In the simpler FBP, which can be solved with either linear or supralinear integration of synaptic inputs, metaplasticity ensures robust, consistently high performance on the task.

We have recently participated in another study with a different goal – whether SPNs can solve the NFBP (Khodadadi et al., 2025). (The NFBP is related to the XOR problem, the canonical minimal example of a linearly non-separable task requiring nonlinear computation beyond a single-layer perceptron.) Khodadadi et al. (2025) uses a different learning rule that incorporates more biological complexity/detail and 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. Although here we study the role of metaplasticity, the results also add some information related to the solution of the NFBP. 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 nonlinearity instead, the SPNs might be able to 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).

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.

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 6–11, 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 so that synapses are flexible (“unlocked”) for learning. 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, 6 and Figure 9 – figure supplement 2. The synaptic weights are initialized uniformly at random from the interval [0.3, 0.35].

Table 1.

Fianlly, the rule uses the amplitudes of [Ca]_NMDA and [Ca]_L-type to trigger plasticity (and metaplasticity, in the scenarios with “thresholded” metaplasticity), implicitly assuming that such values can be remembered by the synapse’s molecular circuitry while dopamine feedback arrives. At least within the stimulation protocol in Fig. 3A, this assumption is justified, since dopamine feedback up to 2 seconds after synaptic stimulation can trigger LTP in experiments (Yagishita et al., 2014).

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 frequencycurrent (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 d_neck = 0.125μm, d_head = 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. 13A, 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. 8. 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 the [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. 13A, B, learning might be prematurely stopped as the [Ca]_NMDA amplitude falls below the increased LTP threshold.

Figure 13.

Because of this, we sought to obtain a monotonic increase of calcium amplitude with increasing plateau voltage (as in Fig. 2C₃), which seems to also occur in experiments where plateaus are evoked (Fig. 2C₂, D₂ in Oikonomou et al. (2012)). Increasing cluster size when evoking NMDA spikes also produces monotonic increases in calcium amplitude, at least for the voltages achieved by NMDA spikes (Figs. 5 and 8 in Losonczy and Magee (2006)). To obtain such monotonic increase, 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 plasmamembrane 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 13–figure supplement 1 (and Fig. 13C). The difference with the diffusion model used throughout the study, which includes VGCCs, is minor (Figure 13–figure supplement 1, cf. Figs. 2C, D). The qualitative and quantitative shapes of the voltage and calcium elevations are very similar (Figure 13–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 13– figure supplement 1B₁), and an obvious difference in the amplitude of [Ca]_NMDA (Figure 13–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. 6. 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. 6 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 30 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 30 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. 7.)

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.

The fraction of NMDAR current that contributes to calcium influx is 17.5%, between the values of 15.9% for receptors with GluN1-GluN2A subunit composition (Table 18 in Traynelis et al. (2010)) and 18.5% (Schneggenburger, 1996).

Glutamate spillover

To produce plateau potentials which have an all-or-none jump in the voltage, i.e. a threshold nonlinearity (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 depends on the synaptic weight w), 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 30 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, 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, making full calcium unbinding and equilibration of the calcium dynamics very long (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. Alternatively, a score of 75% can be achieved by the SPN spiking for both relevant patterns, but being silent for only one irrelevant pattern. Similarly, the threshold score of 87.5% can be reached by being silent for the other irrelevant 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.

Figure supplements

Figure 1—figure supplement 1.

Figure 3—figure supplement 1.

Figure 4—figure supplement 1.

Figure 4—figure supplement 2.

Figure 4—figure supplement 3.

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Figure 4—figure supplement 11.

Figure 7—figure supplement 1.

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Figure 9—figure supplement 2.

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Figure 10—figure supplement 3.

Figure 12—figure supplement 1.

Figure 12—figure supplement 2.

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Figure 12—figure supplement 4.

Figure 13—figure supplement 1.

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.