Figures and data
Synaptic plasticity of the cortico-striatal synapse, the tasks for learning, and the dopamine signals for the two types of patterns in the tasks.
(A) Scheme describing the requirements for synaptic plasticity at the cortico-striatal synapse in dSPNs (adapted from Shen et al. (2008) by adding the calcium thresholds, which are the focus of this study). (B) The FBP and the NFBP, illustrated using features common to the visual system. SPNs might instead receive, e.g. sensoryand motor-related features. (C) Dopamine signals from the midbrain are assumed to arrive in the striatum after every pattern. Dopamine peaks are emitted after the relevant patterns, and dopamine pauses after the irrelevant patterns. (The midbrain and its projections are not explicitly modeled in this study, dopamine is programatically provided to the SPN a certain time after pattern arrival.) This figure is adapted from Khodadadi et al. (2025), licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). Figure 1—figure supplement 1. An example of the NFBP with features relevant for the striatum.
Linear versus supralinear integration of synaptic inputs by SPNs.
(A) Illustrations of the two scenarios of linear integration of distributed synaptic inputs by the soma (A1), and supralinear integration of clustered synaptic inputs in a dendrite (A2). In both scenarios, each synapse is activated with 3 incoming spikes within a window of 30 milliseconds. (B1-B4) Somatic voltage, spine voltage, spine [Ca]NMDA and spine [Ca]L-type evoked in the linear integration scenario (randomly distributed synapses across the dendrites). Line color indicates the number of distributed synapses. In B2-B4, traces from a single spine are shown, which was randomly placed on a randomly chosen dendrite. (C1-C4) Somatic voltage, spine voltage, spine [Ca]NMDA and spine [Ca]L-type evoked in the supralinear integration scenario (synaptic cluster placed on one dendrite, in a 20-micrometer region approximately 120 micrometers away from the soma). Similarly, line color represents the size of the synaptic cluster, which is varied from 1 to 30 synapses. (D1-D4) The amplitude of the somatic voltage, spine voltage, spine [Ca]NMDA and [Ca]L-type compared between the cases of linear and supralinear integration. Results are averages over 20 trials, and in the case of synaptic clusters, over 8 different dendrites with 20 trials per dendrite (clusters located in a 20-micrometer in a dendritic region starting at approximately between 120 micrometers from the soma).
Learning outcome on the FBP and NFBP.
(A) Learning protocol. Patterns are presented in random order, with each pattern consisting of synaptic inputs arriving within a 30 ms window (3 spikes per synapse). Synaptic updates occur during a 50 ms dopamine signal delivered 400 ms after pattern onset; the next pattern is presented 600 ms after pattern onset (see Methods). (B) Illustration of the setups for the FBP and NFBP. (B1) FBP with linear integration: each feature is represented by 15 synapses distributed across the dendrites. (B2) FBP with supralinear integration: 10 synapses per feature, clustered in one dendrite. (B3) NFBP (only supralinear integration): 10 clustered synapses per feature, and clusters are located on two different dendrites. (C) The somatic and dendritic voltages in the FBP with linear (C1) and supralinear (C2) integration, and the NFBP (C3), before and after learning (gray and black traces, respectively), elicited by the relevant and irrelevant pattern(s). (D) The performance on the three tasks. Lines indicate score averaged over consecutive blocks of 20 patterns; shaded areas indicate standard deviation, and dashed lines indicate the threshold scores for solving the FBP (75%) and NFBP (87.5%) (see Methods for what these threshold scores represent). In the FBP with linear (D1) and supralinear (D2) integration, scores are averaged over 50 trials. In (D2) the cluster location is randomized across 8 dendrites in each trial; the cluster is placed in a 20-micrometer dendritic region, approximately 120 micrometers away from the soma. In (D3), the score is averaged over 12 trials per input configuration, divided into two groups (Figure 3–figure supplement 1), resulting in 216 and 156 trials per group, respectively. In each trial, two dendrites are randomly selected from 8 dendrites to place the clusters, at the same distance as in (D2). Figure 3—figure supplement 1. All input configurations used in the NFBP.
The synaptic weights and calcium thresholds during learning of the FBP and NFBP.
(A) Illustrations of the setups for the FBP with linear (A1) and supralinear (A2) integration, and the NFBP (A3) (same as in Fig. 3). (B-D) The evolution of synaptic weights (B), LTP thresholds (C) and LTD thresholds (D) in the FBP with linear (B1-D1) and supralinear (B2-D2) integration, and the NFBP (B3-D3 and B4-D4). In (B) all synaptic weights for the features are shown, while in (C) and (D) only one synapse per feature is chosen to show its calcium threshold (solid lines). Dashed lines in (C2–C4) represent the upper threshold for plasticity ΘLTP. Dots in (C) and (D) represent the amplitudes of [Ca]NMDA and [Ca]L-type, respectively, during pattern presentation. For clarity, the calcium amplitudes for some patterns are omitted. (C) and (D) are shown in larger scale in Figure 4–figure supplement 2, with much more detail in the calcium amplitudes. Arrows show the moment in time when the weakened synapses become low enough for their calcium levels to mostly stay below the calcium thresholds. Figure 4—figure supplement 1. The NFBP cannot be solved with linear integration at the soma. Figure 4—figure supplement 2. Calcium thresholds and calcium amplitudes from Fig. 4 in greater detail. Figure 4—figure supplement 3. The effect of having no upper LTP threshold on supralinear integration. Figure 4—figure supplement 4. A parameter scan of wmax using the rule without an upper LTP threshold. Figure 4—figure supplement 5. The maximal weight wmax prevents unbounded weight growth. Figure 4—figure supplement 6. The effect of the learning rate on learning. Figure 4—figure supplement 7. The effect of a very low learning rate on learning (η = 0.1). Figure 4—figure supplement 8. The effect of a very high learning rate on learning (η = 20). Figure 4—figure supplement 9. The effect of the metaplasticity rate on learning. Figure 4—figure supplement 10. The effect of a low metaplasticity rate on learning (ηθ = 0.2). Figure 4—figure supplement 11. The effect of a high metaplasticity rate on learning (ηθ = 20).
Learning outcome on the FBP and NFBP without metaplasticity in the LTD threshold.
(A) The dendritic and somatic voltage before (gray traces) and after learning (black traces) for the FBP with linear (A1) and supralinear integration (A2) and the NFBP (A3). (B–D) The evolution of synaptic weights (B), LTP thresholds (C) and LTD thresholds (D). (D) shows the fixed LTD thresholds. In (B) all synapses are shown, and in (C, D) only one synapse per feature is used to show its calcium threshold. Dots represent the amplitudes of [Ca]NMDA (C) and [Ca]L-type (D) during pattern presentation, omitting some patterns for clarity. Dashed line in (C2) shows the upper threshold for LTP, ΘLTP. (E) The performance on the FBP with linear (E1) and supralinear integration (E2) and the NFBP (E3) without metaplasticity in the LTD threshold. Performance on the FBP is averaged over 50 trials, and in the NFBP over 12 trials per input configuration (divided into the two groups shown in Figure 3–figure supplement 1), with 216 and 156 trials, respectively).
Learning outcome on the FBP and NFBP without metaplasticity in the LTP threshold.
(A) The dendritic and somatic voltage before (gray traces) and after learning (black traces) for the FBP with linear (A1) and supralinear integration (A2) and the NFBP (A3). (B–D) The evolution of synaptic weights (B), LTP thresholds (C) and LTD thresholds (D) for the synapses in each dendrite. (C) shows the fixed LTP thresholds, and the dashed lines show the upper LTP threshold, ΘLTP. In (B) all synapses are shown, and in (C, D) only one synapse per feature is used to show its calcium threshold. Dots represent the amplitudes of [Ca]NMDA (C) and [Ca]L-type (D) during pattern presentation, omitting some patterns for clarity. Dashed lines in (C2–C4) show the upper threshold for LTP, ΘLTP. (E) The performance on the FBP with linear (E1) and supralinear integration (E2) and the NFBP (E3) without metaplasticity in the LTP threshold. Performance on the FBP is averaged over 50 trials, and in the NFBP over 12 trials per input configuration (divided into the two groups shown in Figure 3–figure supplement 1, with 216 and 156 trials, respectively).
Learning outcome on the NFBP with partial metaplasticity, where the LTP threshold is updated only during LTP, and the LTD threshold is updated only during LTD.
(A) An illustration of the setup for the NFBP (same as in Fig. 3B3). (B) The dendritic and somatic voltage before (gray traces) and after learning (black traces) for all four patterns. (C-E) The evolution of synaptic weights (C), LTP thresholds (D) and LTD thresholds (E) for the synapses in each dendrite. In (C) all synapses are shown, and in (D, E) only one synapse per feature is used to show its calcium threshold. Dots represent the amplitudes of [Ca]NMDA (D) and [Ca]L-type (E) during pattern presentation, omitting some patterns for clarity. Dashed lines in (C2–C4) show the upper threshold for LTP, ΘLTP. (F) The performance on the NFBP with partial metaplasticity, averaged over 12 trials per input configuration (divided into the two groups shown in Figure 3–figure supplement 1, with 216 and 156 trials, respectively). Figure 7—figure supplement 1. The effect of partial metaplasticity on solving the FBP.
Learning outcome on the NFBP without metaplasticity (the LTP and the LTD thresholds are fixed).
(A) An illustration of the setup for the NFBP (same as in Fig. 3B3). (B) The dendritic and somatic voltage before (gray traces) and after learning (black traces) for all four patterns. (C) The evolution of synaptic weights in both dendrites. (D, E) The fixed LTP thresholds (D) and LTD thresholds (E) in each dendrite, with calcium amplitudes during pattern presentation shown with dots (omitting some patterns for clarity). (F) The performance on the NFBP without metaplasticity (same number of trials as in Fig. 7F. Figure 8—figure supplement 1. The effect of having no metaplasticity on solving the FBP.
Reversal learning in the FBP and NFBP cannot be solved with “thresholded” metaplasticity.
(A) The dendritic and somatic voltage before learning (gray traces), before reversal (blue traces) and after reversal (black traces) for the FBP with linear (A1) and supralinear integration (A2) and the NFBP (A3). (B–D) The evolution of synaptic weights (B), LTP thresholds (C) and LTD thresholds (D) for the synapses in each dendrite. In the middle of the simulation, the reward policy is reversed. Dashed lines in (C) show the upper LTP threshold, ΘLTP. In (B) all synapses are shown, and in (C, D) only one synapse per feature is used to show its calcium threshold. Dots represent the amplitudes of [Ca]NMDA (C) and [Ca]L-type (D) during pattern presentation, omitting some patterns for clarity. Dashed lines in (C2–C4) show the upper threshold for LTP, ΘLTP. (E) The performance on reversal learning in the FBP with linear (E1) and supralinear integration (E2) and the NFBP (E3). Performance on the FBP is averaged over 50 trials, and in the NFBP over 12 trials per input configuration (divided into the two groups shown in Figure 3–figure supplement 1), with 216 and 156 trials, respectively). Figure 9—figure supplement 1. “Relaxed” metaplasticity also solves the FBP and NFBP. Figure 9—figure supplement 2. “Relaxed” metaplasticity can overcome initially high calcium thresholds, unlocking synapses and allowing learning.
Reversal learning in the FBP and NFBP is solved when the conditions for metaplasticity are relaxed.
(A) The dendritic and somatic voltage before learning (gray traces), before reversal (blue traces) and after reversal (black traces) for the FBP with linear (A1) and supralinear integration (A2) and the NFBP (A3). (B–D) The evolution of synaptic weights (B), LTP thresholds (C) and LTD thresholds (D) for the synapses in each dendrite. In the middle of the simulation, the reward policy is reversed. Dashed lines in (C) show the upper LTP threshold, ΘLTP. In (B) all synapses are shown, and in (C, D) only one synapse per feature is used to show its calcium threshold. Dots represent the amplitudes of [Ca]NMDA (C) and [Ca]L-type (D) during pattern presentation, omitting some patterns for clarity. (E) The performance on reversal learning in the FBP with linear (E1) and supralinear integration (E2) and the NFBP (E3). Performance on the FBP is averaged over 50 trials. In (E2), due to the thresholds changing slowly, it often occurs that after reward policy reversal, calcium never goes above the thresholds to trigger new learning, thus lowering the performance (blue trace). Increasing the metaplasticity rate to ηθ = 4 can solve this (brown trace). For the NFBP, 12 trials per input configuration are used (divided into the two groups shown in Figure 3–figure supplement 1, with 216 and 156 trials, respectively). Figure 10—figure supplement 1. All input configurations in reversal learning with the NFBP that can store one initially relevant pattern and both irrelevant patterns. Figure 10—figure supplement 2. Relaxed metaplasticity also solves the NFBP in input configurations that can store both (initially) irrelevant patterns. Figure 10—figure supplement 3. Reversal learning in the NFBP on one input configuration that can store both relevant and both irrelevant patterns.
Threshold resetting solves reversal learning.
A signal which resets the thresholds is assumed to arrive from a different brain region detecting the change in reward policy. (A) The dendritic and somatic voltage before learning (gray traces), before reversal (blue traces) and after reversal (black traces) for the FBP with linear (A1) and supralinear integration (A2) and the NFBP (A3). (B–D) The evolution of synaptic weights (B), LTP thresholds (C) and LTD thresholds (D) for the synapses in each dendrite. In the middle of the simulation, the reward policy is reversed. Dashed lines in (C) show the upper LTP threshold, ΘLTP. In (B) all synapses are shown, and in (C, D) only one synapse per feature is used to show its calcium threshold. Dots represent the amplitudes of [Ca]NMDA (C) and [Ca]L-type (D) during pattern presentation, omitting some patterns for clarity. (E) The performance on reversal learning in the FBP with linear (E1) and supralinear integration (E2) and the NFBP (E3). Performance on the FBP is averaged over 50 trials, and in the NFBP over 12 trials per input configuration (divided into the two groups shown in Figure 3–figure supplement 1, with 216 and 156 trials, respectively).
A single plasticity threshold is sufficient to solve the FBP and NFBP.
“Thresholded” metaplasticity is used here. (A) The dendritic and somatic voltage before (gray traces) and after learning (black traces) for the FBP with linear (A1) and supralinear integration (A2) and the NFBP (A3). (B, C) The evolution of synaptic weights (B) and the single calcium threshold (C) for the synapses in each dendrite. Dashed lines in (C) show the upper LTP threshold, ΘLTP. In (B) all synapses are shown, and in (C) only one synapse per feature is used to show its calcium threshold. Dots represent the [Ca] amplitudes during pattern presentation, omitting some patterns for clarity. Dashed lines in (C2–C4) show the upper threshold for LTP, ΘLTP. (D) The performance on the FBP with linear (D1) and supralinear integration (D2) and the NFBP (D3). Performance on the FBP is averaged over 50 trials, and in the NFBP over 12 trials per input configuration (divided into the two groups shown in Figure 3–figure supplement 1, with 216 and 156 trials, respectively). Figure 12—figure supplement 1. A single plasticity threshold is sufficient to solve the FBP and NFBP. when the conditions for metaplasticity are relaxed. Figure 12—figure supplement 2. A single plasticity threshold is sufficient to solve reversal learning when the conditions for metaplasticity are relaxed. Figure 12—figure supplement 3. Learning outcome without metaplasticity in the single threshold scenario (the threshold is fixed). Figure 12—figure supplement 4. [Ca]NMDA and [Ca]L-type are highly correlated and carry essentially the same information.
The parameters in the plasticity rule.
Alternative models of calcium accumulation.
(A) Spine [Ca]NMDA without axial diffusion. [Ca]NMDA is described with a pool model similar to that in Eq. 6 (except that instead of [Ca]L-type, this pool accumulates [Ca]NMDA arising from the NMDA current in Eq. 8). (B) Spine [Ca]NMDA + [Ca]VGCC without axial diffusion. Calcium from the NMDA current and all voltage-gated calcium channels (except L-type) accumulates in a pool model. (C) Spine [Ca]NMDA with axial diffusion. Compared to Fig. 2C3, in this plot only calcium from NMDARs diffuses in the neuron, without calcium from other voltage-gated calcium channels. Figure 13—figure supplement 1. Axial diffusion of calcium from NMDARs only.
The parameters for the axial calcium diffusion model.
Compared to Dorman et al. (2018), we have reduced the concentration of the immobile calcium buffer from 2.5 mM to 0.15 mM and changed the catalytic rate of the PMCA. The diffusion coefficient of Ca2+ is 
The parameters for the pool model for [Ca]L-type.
The parameters in the dual exponential synaptic model.
The parameters in the saturating synapse model.
An example of the NFBP with features relevant for the striatum.
The example is inspired by the tasks used in Bernklau et al. (2024), and consists of pairing a sound frequency with a location where the sound originates from. Two sound frequencies (low and high) and two locations (left and right) were used, with a reward being provided to only two of the feature combinations (low sound to the left and high sound to the right).
All possible input configurations of four features on two dendrites that allow the NFBP to be learned.
They are divided in two groups: those with up to 3 features per dendrite (group 1), and those with 4 features in at least 1 dendrite (group 2). We have made this division because when a dendrite is innervated with all four features (group 2), learning to solve the NFBP depends on the order in which the patterns arrive. In this case, often only one of the relevant patterns is stored by the neuron (the same pattern in both dendrites), and solving the NFBP requires additional mechanisms, such as branch plasticity or inhibition (Legenstein and Maass, 2011; Trpevski et al., 2026).
The NFBP cannot be solved with linear integration at the soma.
Each feature is represented by 15 distributed synapses across the dendrites. Only one relevant pattern is learned by the neuron. (A) The somatic voltage before and after learning. No plateaus are evoked as in Fig. 3B3 because synapses are distributed across the dendrites. After learning, the neuron spikes for only one of the relevant patterns. (B–D) The evolution of synaptic weights (B), LTP thresholds (C) and LTD thresholds (D). In (B) synaptic weights for all synapses are shown, while in (C) and (D) only one synapse per feature is chosen to show its calcium threshold. Dots represent the amplitudes of [Ca]NMDA and [Ca]L-type, respectively, during pattern presentation. In this simulation, only the “yellow banana” is stored by the neuron’s distributed synapses (B). Because no plateaus are evoked, [Ca]NMDA never reaches the upper threshold for LTP, ΘLTP (C), and as a result, synapses are strengthened close to the maximal value wmax, as in the FBP with linear integration (B, cf. Fig. 4B1). (D) Performance on the NFBP averaged over 50 trials. The dashed lines mark the threshold scores of 75% and 87.5%.
The calcium thresholds and calcium amplitudes from Fig. 4 shown in larger panels with greater detail.
(A) LTP thresholds for the FBP with distributed (A1) and clustered (A2) synapses, and for the NFBP (A3, A4). (B) LTD thresholds for the FBP with distributed (B1) and clustered (B2) synapses, and for the NFBP (B3, B4). In (B3), regions with calcium amplitudes of interest are highlighted (the main text refers to these regions when describing the dynamics of the weights, thresholds, and calcium concentrations). The same regions exist in all panels, although they are not highlighted.
The effect of no upper threshold ΘLTP on supralinear integration.
(A) The somatic and dendritic voltages before and after learning in the FBP (A1) and NFBP (A2). Not having an upper threshold causes somatic spiking for the irrelevant patterns after learning in the NFBP, but not the FBP. (B-D) The evolution of synaptic weights (B), LTP thresholds (C) and LTD thresholds (D) in the FBP (B1-D1) and the NFBP (B2-D2 and B3-D3). Strengthened synapses saturate at their maximal levels, also driving more weakening in the weakened synapses. Having no upper threshold does not affect the thresholds of the strengthened synapses, but causes the thresholds of the weakened synapses to stabilize at higher levels. (E) Performance on the FBP (E1) and NFBP (E2). The same number of trials as in Fig. 3D is used.
A parameter scan of wmax using the rule without an upper LTP threshold.
(A) The performance on the FBP with linear (A) and supralinear integration (B) and on the NFBP (C) is shown when the maximal weight is varied from 
The maximal weight wmax prevents unbounded weight growth.
In these simulations, no wmax was used, causing the weights to increase until calcium concentration saturates in the spines. (A) The somatic and dendritic voltages in the FBP with linear (A1) and supralinear (A2) integration before and after learning (gray and black traces, respectively), elicited by the relevant and irrelevant pattern. wmax ensures firing rate stability in the FBP with linear integration – without it, the neuron enters into a depolarization block after learning. In the FBP with supralinear integration, the plateaus ensure firing rate stability – regardless of how high the synaptic weights are, a plateau’s voltage is limited by the NMDAR reversal potential, providing dynamic range compression (see also Figs. 3E, 4D and 12 in Oikonomou et al. (2012)). (B) The evolution of the weights in the FBP with linear (B1) and supralinear (B2) integration. (C, D) The LTP (C) and LTD threshold (D) shown for a single synapse per feature in the FBP with linear (C1, D1) and supralinear (C2, D2) integration. No upper threshold for LTP was used in the FBP with supralinear integration in order to showcase the effect of not having a maximal weight cap, wmax. A modified version of the spillover model was used here in which the synaptic weight does not prolong the plateaus’ duration (otherwise, an increasingly long inter-pattern interval would have been needed to allow the plateau to finish, significantly prolonging the simulation.)
The effect of the learning rate η on learning.
(A) The average score on the FBP and NFBP for three values of the learning rate, η ∈ {0.4, 0.85, 1.7} (the number of trials is the same as in Fig. 3D). Increasing the learning rate causes faster learning of the tasks (less pattern presentations are needed). (B) The score at the end of learning varies little with η, except in group 2 for the NFBP, where a higher learning rate increases performance, so that one of the relevant patterns is remembered. (C) The standard deviation at the end of learning also varies little with η (there is a 5% decrease for the NFBP, group 1 only). (D) The speed of learning measured by the number of patterns, Ns, needed to reach the score threshold for solving each task (the dashed line in (A)). The speed of learning to solve the tasks is affected by the learning rate η, as is also seen in (A). Increasing the learning rate speeds up learning.
The effect of a very low learning rate on learning (η = 0.1).
(A) The somatic and dendritic voltages in the FBP with linear (A1) and supralinear (A2) integration, and the NFBP (A3), before and after learning (gray and black traces, respectively), elicited by the relevant and irrelevant pattern(s). (B-D) The evolution of synaptic weights (B), LTP thresholds (C) and LTD thresholds (D) in the FBP with linear (B1-D1) and supralinear (B2-D2) integration, and the NFBP (B3-D3 and B4-D4). The low value of η cuses small weight updates, resulting in less weakening than is necessary to solve the NFBP (E3), and significantly prolonging the learning of the FBP with linear integration (E1). In (B) all synaptic weights for the features are shown, while in (C) and (D) only one synapse per feature is chosen to show its calcium threshold (solid lines). Dots in (C) and (D) represent the amplitudes of [Ca]NMDA and [Ca]L-type, respectively, during pattern presentation. Dashed lines in (C2–C4) show the upper threshold for LTP, ΘLTP. (E) The performance on the FBP with linear (E1) and supralinear (E2) integration, and on the NFBP (E3). In (E1) the score is averaged over 10 trials and (E2) the score is averaged over 50 trials. In (E3) the score is averaged over 4 trials per input configuration, divided into two groups (Figure 3–figure supplement 1), resulting in 72 and 52 trials per group, respectively.
The effect of a very high learning rate on learning (η = 20).
(A) The somatic and dendritic voltages in the FBP with linear (A1) and supralinear (A2) integration, and the NFBP (A3), before and after learning (gray and black traces, respectively), elicited by the relevant and irrelevant pattern(s). (B-D) The evolution of synaptic weights (B), LTP thresholds (C) and LTD thresholds (D) in the FBP with linear (B1-D1) and supralinear (B2-D2) integration, and the NFBP (B3-D3 and B4-D4). Large updates occur in the weights due to the high value of η, which in the NFBP cause calcium to fall below the thresholds and prematurely stop learning, without storing any pattern. The large weight updates even cause a plateau very early during learning, for the first irrelevant pattern (“yellow strawberry” in A3). In (B) all synaptic weights for the features are shown, while in (C) and (D) only one synapse per feature is chosen to show its calcium threshold (solid lines). Dots in (C) and (D) represent the amplitudes of [Ca]NMDA and [Ca]L-type, respectively, during pattern presentation. Dashed lines in (C2–C4) show the upper threshold for LTP, ΘLTP. (E) The performance on the FBP with linear (E1) and supralinear (E2) integration, and on the NFBP (E3). In (E1) and (E2) the score is averaged over 10 trials. In (E3) the score is averaged over 8 trials per input configuration, divided into two groups (Figure 3–figure supplement 1), resulting in 144 and 104 trials per group, respectively.
The effect of the metaplasticity rate ηθ on learning.
(A) The average score on the FBP and NFBP for four values of the metaplasticity rate, ηθ ∈ {1, 2, 3, 4} (the standard deviation is ommited for figure clarity, but the number of trials is the same as in Fig. 3D). There is little effect of the metaplasticity rate on the final score (B), the final standard deviation of the score (C) and the speed of learning (D). (B) The score at the end of learning varies little with ηθ (only slightly for the NFBP). (C) The standard deviation at the end of learning also varies little with ηθ (up to 5% increase for the NFBP only). (D) The speed of learning measured by the number of patterns, Ns, needed to reach the score threshold for solving each task (the dashed line in (A)). The speed of learning to solve the FBP varies very little with ηθ, while that for solving the NFBP only shows a large increase between ηθ = 1 and ηθ = 2.
The effect of a low metaplasticity rate on learning (ηθ = 0.2).
(A) The somatic and dendritic voltages in the FBP with linear (A1) and supralinear (A2) integration, and the NFBP (A3), before and after learning (gray and black traces, respectively), elicited by the relevant and irrelevant pattern(s). (B) The evolution of the weights in the FBP with linear (B1) and supralinear (B2) integration, and the NFBP (B3, B4). When ηθ is very low, the weights take a longer time to stabilize (note that the simulation on the NFBP is run three times longer). (C, D) The LTP (C) and LTD threshold (D) shown for a single synapse per feature in the FBP with linear (C1, D1) and supralinear (C2, D2) integration, and the NFBP (C3,4, D3,4). Because of the low metaplasticity rate, none of the thresholds reach the maximal calcium levels within the length of these simulations. (E) The performance on the FBP with linear (E1) and supralinear (E2) integration, and on the NFBP (E3). The low value of ηθ does not affect learning in the FBP (cf. Fig. 3D1, D2) but prolongs learning in the NFBP (cf. Fig. 3D3). In (E1) and (E2) the performance is averaged over 10 trials, and in (E3) over 4 trials per input configuration, which are divided into the two groups shown in Figure 3–figure supplement 1 (with 72 and 52 trials in the two groups, respectively).
The effect of a low metaplasticity rate on learning (ηθ = 20).
(A) The somatic and dendritic voltages in the FBP with linear (A1) and supralinear (A2) integration, and the NFBP (A3), before and after learning (gray and black traces, respectively), elicited by the relevant and irrelevant pattern(s). (B) The evolution of the weights in the FBP with linear (B1) and supralinear (B2) integration, and the NFBP (B3, B4). (C, D) The LTP (C) and LTD threshold (D) shown for a single synapse per feature in the FBP with linear (C1, D1) and supralinear (C2, D2) integration, and the NFBP (C3,4, D3,4). When ηθ is very high, the thresholds quickly reach their respective calcium levels, and stabilize the weights more quickly. As a result, the weights are modified less with respect to their initial values. (E) The performance on the FBP with linear (E1) and supralinear (E2) integration, and on the NFBP (E3). The high value of ηθ causes more variability in the performance for the FBP with linear integration because weights are not strengthened enough. It also lowers the performance on the NFBP because weakened clusters are not weakened enough. In (E1) and (E2) the performance is averaged over 10 trials, and in (E3) over 4 trials per input configuration, which are divided into the two groups shown in Figure 3–figure supplement 1 (with 72 and 52 trials in the two groups, respectively).
Learning outcome on the FBP with linear (A1-E1) and supralinear integration (A2-E2) with partial metaplasticity.
(A) The somatic and dendritic voltage evoked by both patterns before and after learning. (B-D) The evolution of synaptic weights (B), LTP thresholds (C) and LTD thresholds (D) during learning. In (B) all synaptic weights are shown, and in (C, D) the thresholds for only one synapse per feature are shown. (E) The performance on the FBP, averaged over 50 trials.
Learning outcome on the FBP with linear (A1-E1) and supralinear integration (A2-E2) without metaplasticity.
(A) The somatic and dendritic voltage evoked by both patterns before and after learning. (B) The evolution of synaptic weights during learning. (C, D) The LTP (C) and LTD thresholds (D) are fixed. In (B) all synaptic weights are shown, and in (C, D) the thresholds for only one synapse per feature are shown. (E) The performance on the FBP, averaged over 50 trials.
Relaxed metaplasticity also solves the FBP and NFBP.
(A) The somatic and dendritic voltage before (gray traces) and after learning (black traces) for the FBP with linear (A1) and supralinear integration (A2) and the NFBP (A3).(B–D) The evolution of synaptic weights (B), LTP thresholds (C) and LTD thresholds (D) during learning. The thresholds for the shared features that need to be strengthened show large fluctuations in the cases of supralinear integration (C2–C4 and D2–D4). However, this does not influence the performance on the tasks. In (B) all synaptic weights are shown, and in (C, D) the thresholds for only one synapse per feature are shown. (E) The performance on the FBP with linear (E1) and supralinear integration (E2) and the NFBP (E3). Performance on the FBP is averaged over 50 trials, and in the NFBP over 12 trials per input configuration (divided into the two groups shown in Figure 3–figure supplement 1, with 216 and 156 trials, respectively).
“Relaxed” metaplasticity can overcome initially high calcium thresholds, unlocking synapses and allowing learning.
Only the NFBP is used as a demonstration. The somatic and dendritic voltage before (gray traces) and after learning (black traces) for “thresholded” (A1) and “relaxed” metaplasticity (A2). (B–D) The evolution of synaptic weights (B), LTP thresholds (C) and LTD thresholds (D) during learning. In (B) all synaptic weights are shown, and in (C, D) the thresholds for only one synapse per feature are shown. Dots in (C, D) represent calcium amplitudes evoked by the patterns. With “thresholded” metaplastiity, initializing thresholds to high values leaves the synapses locked (B1, B2), as calcium amplitudes always remain below the thresholds (C1,2, D1,2). The NFBP cannot be solved in this case (A1). On the other hand, with “relaxed” metaplasticity, the initially high thresholds adapt (C3,4, D3,4), thus unlocking the synapses and allowing learning (B3, B4). The NFBP is solved in this case (A2).
All possible input configurations of four features on two dendrites where one initially relevant pattern and both irrelevant patterns can be stored through in the simulations with reversal learning.
They are divided in two groups: those with up to 3 features per dendrite (group 1), and those with 4 features in at least 1 dendrite (group 2). This division is made because when a dendrite is innervated with all four features (group 2), learning to solve the NFBP depends on the order in which the patterns arrive and solving the NFBP requires additional mechanisms, such as branch plasticity or inhibition (Legenstein and Maass, 2011; Trpevski et al., 2026).
Relaxed metaplasticity also solves the NFBP in input configurations that can store both (initially) irrelevant patterns.
(A) The dendritic and somatic voltage before learning (gray traces), before reversal (blue traces) and after reversal (black traces). (B–D) The evolution of synaptic weights (B), LTP thresholds (C) and LTD thresholds (D) during learning. In (B) all synaptic weights are shown, and in (C, D) the thresholds for only one synapse per feature are shown. (E) The performance on the NFBP in the input configurations that can store one relevant pattern and both irrelevant patterns (shown in Figure 10–figure supplement 1). 12 trials per input configuration were used, divided into the two groups shown in Figure 3–figure supplement 1, with 216 and 156 trials, respectively).
Reversal learning in the NFBP on one input configuration that can store both relevant and both irrelevant patterns.
(A) An input configuration that allows the initially relevant patterns ‘red strawberry’ and ‘yellow banana’ to be stored in dendrite 1 and 2, respectively, as well as, the initially irrelevant patterns ‘yellow strawberry’ and ‘red banana’ to be stored in dendrites 1 and 2 after reversal of the reward policy. (B) The somatic and dendritic voltage before learning (gray traces), before reward policy reversal (blue traces) and after reversal (black traces).(C–E) The evolution of synaptic weights (C), LTP thresholds (D) and LTD thresholds (E) during learning. In (C) all synaptic weights are shown, and in (D, E) the thresholds for only one synapse per feature are shown. (F) The performance on the NFBP in all four input configurations in Figure 3– figure supplement 1 that can store both relevant and irrelevant patterns. 32 trials were performed for each input configuration, for a total of 128 trials.
A single plasticity threshold is sufficient to solve the FBP and NFBP when the conditions for metaplasticity are relaxed.
(A) The dendritic and somatic voltage before (gray traces) and after learning (black traces) for the FBP with linear (A1) and supralinear integration (A2) and the NFBP (A3). (B, C) The evolution of synaptic weights (B) and the single calcium threshold (C) for the synapses in each dendrite. Dashed lines in (C) show the upper LTP threshold, ΘLTP. In (B) all synapses are shown, and in (C) only one synapse per feature is used to show its calcium threshold. Dots represent the [Ca] amplitudes during pattern presentation, omitting some patterns for clarity. (E) The performance on the FBP with linear (E1) and supralinear integration (E2) and the NFBP (E3). Performance on the FBP is averaged over 50 trials, and in the NFBP over 12 trials per input configuration (divided into the two groups shown in Figure 3–figure supplement 1, with 216 and 156 trials, respectively).
A single plasticity threshold is sufficient to solve reversal learning when the conditions for metaplasticity are relaxed.
(A) The dendritic and somatic voltage before learning (gray traces), before reward policy reversal (blue traces) and after reversal (black traces) for the FBP with linear (A1) and supralinear integration (A2) and the NFBP (A3). (B, C) The evolution of synaptic weights (B) and the single calcium threshold (C) for the synapses in each dendrite. Dashed lines in (C) show the upper LTP threshold, ΘLTP. In (B) all synapses are shown, and in (C) only one synapse per feature is used to show its calcium threshold. Dots represent the [Ca] amplitudes during pattern presentation, omitting some patterns for clarity. (D) The performance on the FBP with linear (E1) and supralinear integration (E2) and the NFBP (E3, E4). Performance on the FBP is averaged over 50 trials, and in the NFBP over 12 trials per input configuration (divided into the two groups shown in Figure 3–figure supplement 1, with 216 and 156 trials, respectively).
Learning outcome without metaplasticity in the single threshold scenario (the threshold is fixed).
Similarly to when two thresholds are used, only the FBP is solved with reduced performance. (A) The dendritic and somatic voltage before (gray) and after learning (black) for the FBP with linear (A1) and supralinear integration (A2) and the NFBP (A3). (B) The evolution of synaptic weights. (C) The single calcium threshold is fixed. Dashed lines show the upper LTP threshold, ΘLTP, and only one synapse per feature is used to show its calcium threshold. Dots represent the [Ca] amplitudes during pattern presentation, omitting some patterns for clarity. (D) The performance on the FBP with linear (D1) and supralinear integration (D2) and the NFBP (D3, E4). Performance on the FBP is averaged over 10 trials, and in the NFBP over 5 trials per input configuration (divided into the two groups shown in Figure 3–figure supplement 1, with 216 and 156 trials, respectively).
[Ca]NMDA and [Ca]L-type are highly correlated and carry essentially the same information.
The analysis is done on calcium signals from one learning simulation of the NFBP with “thresholded” metaplasticity. (A) Cross-correlation of the normalized [Ca]NMDA and [Ca]L-type in a single synapse, averaged over all clustered synapses. (B) Zoom-in on the crosscorrelation function shows that the peak is centered sligthly off 0, at around -35 ms. (C) Histogram of the peak cross-correlation lag suggests that [Ca]NMDA precedes [Ca]L-type by about 35 ms. (D) A scatter plot of the amplitudes of [Ca]NMDA and [Ca]L-type (whose values are used to check if the calcium thresholds are crossed in the learning rule), with a fitted linear regression model. The coefficient of determination is R2 ≈ 0.98141, so in the linear regression model, max[Ca]NMDA explains 98.14% of the variance in max[Ca]L-type. Also, Pearson’s correlation coefficient for the calcium amplitudes is r ≈ 0.99066, indicating that they are highly correlated. (Note that in this case, r2 = R2). (E) A scatter plot of the entire [Ca]NMDA and [Ca]L-type time-courses in all clustered synapses. It shows a circular structure, in agreement with the fact that [Ca]NMDA precedes [Ca]L-type. (F) A scatter plot of the entire [Ca]NMDA and [Ca]L-type time-courses in all clustered synapses where [Ca]L-type is shifted by 35 ms in order to be aligned with [Ca]NMDA. The time-courses also have a pronounced linear relationship. (G) Principal component analysis of the normalized [Ca]NMDA and [Ca]L-type time-courses ([Ca]L-type was not shifted by any amount). PC1 explains around 94.26 % of the variance in the two calcium signals, suggesting that the joint calcium dynamics are strongly dominated by a onedimensional variable (the membrane voltage). (PC2 explains 5.74 % of the variance.)
Voltage and calcium elevations when only [Ca]NMDA diffuses axially through the SPN.
(A1-C4) Somatic voltage, spine voltage, spine [Ca]NMDA and spine [Ca]L-type evoked in the supralinear integration scenario, to be compared with Fig. 2C (synaptic cluster placed on one dendrite, in a 20-micrometer region approximately 120 micrometers away from the soma). Line color represents the size of the synaptic cluster, which is varied from 1 to 30 synapses. (B1-B4) The amplitude of the somatic voltage, spine voltage, spine [Ca]NMDA and [Ca]L-type compared between the cases of only [Ca]NMDA diffusing and [Ca]NMDA + [Ca]VGCC diffusing (the latter is replfrom Fig. 2D). Results are averages over 20 trials, and in the case of synaptic clusters, over 8 different dendrites (clusters located in a 20-micrometer in a dendritic region starting at approximately between 120 micrometers from the soma).
