Local, calcium- and reward-based synaptic learning rule that enhances dendritic nonlinearities can solve the nonlinear feature binding problem
- Science for Life Laboratory, Department of Computer Science, KTH Royal Institute of Technology, Sweden
- Department of Neuroscience, Karolinska Institutet, Sweden
Figures
Figure 1
Learning mechanisms in direct-pathway striatal projection neurons (dSPNs) for the nonlinear feature binding problem (NFBP).
(A) Inputs and assumed supralinearity that could solve the NFBP: The NFBP is represented with an example from visual feature binding. In the simplest form of the NFBP, a stimulus has two features, here shape and form, each with two possible values: strawberry and banana, and red and yellow, respectively. In the NFBP, the neuron should learn to respond by spiking to two of the feature combinations, representing the relevant stimuli (red strawberry and yellow banana), while remaining silent for the other two feature combinations which represent the irrelevant stimuli (yellow strawberry and red banana). Assuming that each feature is represented with locally clustered synapses, a solution of the NFBP can be achieved when the co-active clusters on a single dendrite, corresponding to a relevant stimulus, evoke a plateau potential, thus superlinearly exciting the soma. Conversely, co-activation of synaptic clusters for the irrelevant combinations should not evoke plateau potentials. (B) Synaptic clustering in dendrites: Illustration of how synaptic plasticity in SPNs may contribute to solving the NFBP for a pre-existing arrangement of synaptic clusters on two dendrites. A plasticity rule that strengthens only synaptic clusters representing relevant feature combinations so that they produce robust supralinear responses, while weakening synapses activated by irrelevant feature combinations, could solve the NFBP. (C) Dopamine (Da) feedback: Dopaminergic inputs from the midbrain to the striatum (Str) guide the learning process, differentiating between positive feedback for relevant stimuli and negative feedback for irrelevant stimuli. Positive feedback, represented by a dopamine peak, is necessary for long-term potentiation (LTP), and negative feedback, represented by a dopamine pause, is necessary for long-term depression (LTD). (D) Signaling pathways underlying synaptic plasticity in dSPNs: illustrations of signaling components at the corticostriatal synapse that modify synaptic strength (redrawn from Shen et al., 2008). NMDA calcium influx, followed by stimulation of D1 dopamine receptors (D1Rs), triggers LTP (while inhibiting the LTD cascade). L-type calcium influx and activation of metabotropic glutamate receptors (mGluRs) when D1Rs are free of dopamine triggers LTD (while counteracting the LTP cascade).
Figure 2
Characterization of dendritic plateau potentials in the model.
(A) Somatic voltage, spine voltage, NMDA calcium, and L-type calcium evoked by a cluster varying in size from 1 to 20 synapses. A plateau potential is evoked when glutamate spillover is activated, here triggered when 8 synapses with a weight of 0.25 each are coactivated (corresponding to the ‘baseline’ weights in C). (B) Schematic of the neuron morphology with an arrow indicating an example location of a stimulated dendritic branch. (C) The mean maximal amplitude, with standard deviation shown in bars, of the measures in A averaged over 10 different dendrites and 10 trials per dendrite (for a total of n = 100 trials). The curves represent clusters with different synaptic weights (conductances): baseline – 0.25 (0.625 nS), strengthened – 0.4 (1 nS), and weakened – 0.2 (0.5 nS). Synaptic background noise is used in all simulations to elevate the membrane potential to ranges seen in vivo (Reig and Silberberg, 2014).
Figure 3 with 2 supplements
Example of setup and learning-induced synaptic plasticity.
(A) Input configuration. The panel shows how the four stimulus features are distributed across two dendrites (d1, d2). Each dendrite contains pre-existing synaptic clusters for three features (black circles) and distributed, feature-unspecific synapses (shown in gray). (B) Stimulation timeline and plasticity schematic. The top diagram depicts a 20 ms stimulus followed by a 50 ms dopaminergic feedback pulse delivered with a 300 ms delay, only if the neuron spikes. Successive stimuli are separated by 800 ms to let calcium return to baseline. Lower panels: a dopamine peak (left) gates long-term potentiation (LTP) – synapses whose falls within the bell-shaped window are potentiated (, green) while the kernel midpoint shifts downward (↓, red); a dopamine pause (right) gates long-term depression (LTD) – the degree of depression scales with (, red) and shifts upward (↑, green). (C, D) Evolution of synaptic conductances during learning (top row) and examples of peak Ca2+ levels (black dots) alongside kernel dynamics in single synapses (three lower rows). Panels in (C) depict clustered synapses in dendrite 1 (d1), where ‘yellow’ and ‘banana’ generally undergo LTP, while ‘red’ undergoes LTD. See Figure 3—video 1 for an animation of panel C showing the first 400 training stimuli of the learning sequence. Panels in (D) show distributed, feature-unspecific synapses. Among these, Example 1 and Example 3 traces represent synapses that are weakened, while Example 2 trace exemplifies a synapse near one of the clusters that, by chance, achieves a sufficiently high local NMDA calcium level for LTP to dominate. The initial synaptic conductances are drawn from a normal distribution with a mean of 0.625 nS and a standard deviation of 0.125 nS. The solid line represents the midpoint of the kernels, where LTP is strongest. 'Max' indicates the peak NMDA calcium during a single stimulus. The darker shaded regions represent the LTP kernel, and the lighter shaded ones show the wider metaplasticity kernel. (E) Example voltage in the soma and the middle of dendrite 1 (d1) and dendrite 2 (d2) before and after learning. Each dendrite here stops responding to the respective irrelevant stimuli during learning.
Figure 3—figure supplement 1
Learning-induced synaptic plasticity with metaplasticity turned off.
(A) Example voltage traces recorded at the soma and in two dendrites before and after learning. Without metaplasticity, no significant learning occurs – every stimulus triggers spikes (50% performance). (B, C) Evolution of synaptic conductances throughout learning (top row) and characteristic examples of peak calcium levels (dots), along with the constant kernels in single synapses (shaded areas, bottom rows). The same synaptic arrangement and initial synaptic weights as in Figure 3A are used (for both clustered and feature-unspecific synapses). (B) illustrates changes in clustered synapses on dendrite 1 (d1). (C) shows distributed, feature-unspecific synapses. The final outcome depends on the initial synaptic weights. Synapses whose calcium levels are within or above the long-term potentiation (LTP) region cannot be weakened, because any long-term depression (LTD) during irrelevant stimuli is counteracted by LTP occurring for the relevant stimuli. The calcium levels remain ‘trapped’ within the LTP kernel, and the synaptic weights ‘zig-zag’ around a steady-state level (seen in the example clustered ‘red’, ‘yellow’, and ‘banana’ synapse, as well as for Example 2 and Example 3 feature-unspecific synapses). The synapses with small initial weights, whose calcium levels are situated below the LTP kernel, are weakened (Example 1 synapse). The solid lines within the LTP kernels represent the midpoint of the kernels, where LTP is strongest. In summary, without metaplasticity, learning of the nonlinear feature binding problem (NFBP) is difficult if initial synaptic weights are not optimally tuned with regard to the postsynaptic calcium concentration. Metaplasticity allows the synapses to fall in and out of the LTP kernel range. They can be strengthened as long as their calcium levels are within the LTP kernel, are stabilized when they are above the LTP kernel, and can be weakened when they are below the LTP kernel.
Figure 3—video 1
Animated rendering of the first 400 training stimuli (~400 epochs) showing frame-by-frame changes in conductance and calcium for the clustered synapses shown in panel C.
Figure 4
Impact of input configurations and synaptic cluster locations on nonlinear feature binding problem (NFBP) learning performance.
(A) Illustration of creating different input configurations where two, three (light blue circles), or four features (deep red circles) are placed in two dendritic locations. (B) Performance trajectories for all 31 unique input configurations over training epochs, shown for one particular pair of dendrites. Each colored line represents the learning performance of a single configuration, with blue traces indicating setups where no dendrite receives more than three features and red traces representing setups where at least one dendrite contains all four features. The black line and gray shaded area represent the mean and standard deviation over all trials (n = 31). The inset shows the number of configurations in each group that reached the NFBP learning criterion: 18/18 for the blue group and 5/13 for the deep red group. (C) Performance on the last 160 stimuli for the two groups of configurations: (C1) setups where each dendrite has at most three features and (C2) setups where at least one dendrite contains all four features. (D, E) Performance in a three-feature configuration as a function of cluster location. (D) illustrates the distribution of synaptic clusters across dendritic locations, and (E) shows performance over the last 160 stimulus presentations as a function of somatic distance for these configurations. A total of 60 unique dendritic arrangements were tested, with synapse clusters randomly assigned to different dendritic locations. The solid line represents a quadratic fit to the data.
Figure 5 with 1 supplement
Effects of inhibitory plasticity on performance.
(A) Dendritic input configuration with inhibitory synapses added. The setup of the excitatory and feature-unspecific synapses is the same as in Figure 4A. Plastic inhibitory synaptic connections for each of the four features are added in both dendrites, with one synapse per feature. To the right is a schematic of the inhibitory plasticity rule. Active inhibitory synapses strengthen at lower calcium levels and weaken when calcium is high, whereas inactive inhibitory synapses follow the opposite pattern. (B) displays average performance for 31 unique input configurations of 2, 3, or 4 features on two dendrites as a comparison between the setup with (orange) and without inhibitory plasticity (blue). Shaded areas show standard deviation (n = 31). The bar plot inset shows the number of configurations, with 4 features in at least one dendrite, that successfully solved the nonlinear feature binding problem (NFBP) with and without inhibitory plasticity. (C) Performance over the last 160 stimulus presentations as a function of somatic distance of the synaptic clusters for the input configuration in Figure 4D, with added inhibition. A total of 60 unique dendritic arrangements were tested, with synapse clusters randomly assigned to different dendritic locations. The solid line represents a quadratic fit to the data. The dashed line is the corresponding quadratic fit from Figure 4E. (D) Peak voltage-gated calcium (left panel, dots) and plasticity threshold dynamics (lines), in dendrite 1, over training epochs. The upper threshold () moves toward peak calcium levels while the lower threshold () moves toward the next highest level. (Right) Inhibitory synaptic conductances for the synapses in dendrite 1. Strengthened inhibitory synapses prevent excitation by the corresponding excitatory features (here ‘red’ and ‘strawberry’), while weakened inhibitory synapses allow excitation by their corresponding features (here ‘yellow’ and ‘banana’). (E) Excitatory synaptic conductances (top) and calcium levels and kernel dynamics (below) over learning. The specific conductances in the top panel corresponding to the kernel dynamics and calcium levels in the bottom panels are indicated with arrows. Note that kernel and calcium dynamics for the example ‘yellow’ synapse refer to the only ‘yellow’ synapse in the top panel which initially weakens and is later strengthened. The solid line shows the midpoint of the long-term potentiation (LTP) kernels. Max refers to the peak NMDA calcium during a single stimulus. The darker shaded regions represent the LTP kernel, and the lighter shaded ones show the wider metaplasticity kernel.
Figure 5—figure supplement 1
Learning with and without inhibitory plasticity.
(A) The figure illustrates the impact of inhibitory plasticity on learning dynamics in two distal dendrites (d1 and d2). The top panels depict somatic voltage, while the middle and bottom panels show dendritic voltages. The left panels show voltage in the middle of the learning process (approximately 480 training epochs) with and without inhibitory plasticity. The right panels show voltage at the end of the learning process (approximately 960 training epochs). The dashed black line corresponds to the scenario without inhibitory plasticity, and the solid red line indicates the presence of inhibitory plasticity. In the middle of learning, inhibition prevents spiking for the irrelevant stimuli, thus leading to less long-term depression (LTD). This preserves synaptic strengths in both the clustered and feature-unspecific synapses, allowing the neuron to spike for the relevant stimuli at the end of learning. The overall decrease in unspecific input was 39% without inhibitory plasticity and 35% with inhibitory plasticity. (B) The effect of inhibitory plasticity on learning in a dendrite receiving all four features. The left panel shows excitatory synaptic weights in d1 without inhibitory plasticity, where all features weaken over time due to competition. The right panel illustrates excitatory synapses in the presence of inhibitory plasticity, where learning is better regulated.
Figure 6 with 1 supplement
Performance analysis of learning using distributed synaptic inputs.
(A) Example illustration of synaptic distribution before (top) and after learning (bottom) of the 200 excitatory and 60 inhibitory inputs. S – strawberry, Y – yellow, B – banana, R – red, U – feature-unspecific inputs. Bordered markers (circles and diamonds) indicate synapses whose conductances are shown in (C). (B) Performance over training epochs with and without inhibitory plasticity. (Left) Mean performance of the setups with and without inhibitory plasticity. Shaded areas show standard deviation. (Right) Individual traces for the setup with inhibitory plasticity. Each of these individual trials (n = 31 in total) uses a unique random distribution of synapses. (C) Example of summed synaptic conductances (left) and voltage (right) in the soma, and four example dendrites (d1–d4) of the synaptic distribution in (A) (corresponding to a trial where the nonlinear feature binding problem (NFBP) is successfully solved). The sums of both excitatory (left) and inhibitory (right) inputs are shown.
Figure 6—figure supplement 1
Other learning paradigms.
(A) Learning of 5×5 and 3×3 feature combination tasks with randomly distributed synapses over 30 dendrites. The neuron is trained to spike for black-marked combinations and remain silent for white-marked ones. For the 25-feature (5×5) combination, the model used 500 excitatory synapses (50 per feature) with initial conductances of 1.125 ± 0.25 nS, along with 100 inhibitory synapses (10 per feature) with initial conductances of 0.1 ± 0.01 nS. For the 9-feature (3×3) combination, the model used 312 excitatory synapses (52 per feature) and 96 inhibitory synapses (16 per feature), with the same initial weights as in the 5×5 case. (B) Learning of a 2×2 linear feature combination task with randomly distributed synapses. Each feature is represented by 75 synapses distributed across 30 dendrites (300 total synapses), with an initial synaptic weight of 0.375 ± 0.25 nS. The top row shows the feature combinations used, followed by the performance across 30 different synaptic distributions. The bottom row illustrates somatic voltage traces before and after learning, where the neuron starts from a silent state (red trace) and learns to spike through metaplasticity and dopamine-driven weight updates. Additionally, in the subthreshold learning paradigm, dopamine was given without spiking for expected firings and suppressed when unexpected spiking occurred. (C) Evolution of synaptic conductances (left column) and calcium dynamics with metaplasticity activity (right column) during learning for each feature (X1, X2, Y1, Y2). The left panels display the weight adjustments of all synapses corresponding to each feature, with one synapse per feature bolded to highlight its individual trajectory. The right panels show the corresponding calcium traces and kernel dynamics for the bolded synapse. The solid line represents the midpoint of the kernels where long-term potentiation (LTP) is strongest. ‘Max’ indicates the peak NMDA calcium level during a single stimulus. The darker shaded regions represent the LTP kernel, and the lighter shaded ones show the wider metaplasticity kernel.
Figure 7
Synaptic plasticity rules: calcium and dopamine interactions in synaptic weight modification.
(A) Synaptic weight updates following a dopamine peak. (Top) The long-term potentiation (LTP) kernel is a bell-shaped curve describing the amount of weight increase, which happens over a region of . (Bottom) A wider bell-shaped kernel, i.e., the metaplasticity kernel, determines how the LTP kernel is shifted along the calcium level () axis following a peak in dopamine. (B) Synaptic weight updates following a dopamine pause. (Top) The long-term depression (LTD) plasticity kernel. The LTD threshold is constant and set at 70 nM. (Bottom) Metaplasticity describing how the LTP kernel shifts along the calcium axis following a dopamine pause. (C) A schematic of how the LTP kernel is shifted following a dopamine peak (left) and pause (right), together with examples of NMDA calcium levels before and after the shift. The NMDA calcium levels change following the potentiation or suppression of the synapse (illustrated with the red circle jumping to the blue circle). The LTP kernel also moves as indicated by the arrows and colors following a peak or a pause, respectively. (D) Illustration of the inhibitory plasticity rule. (Top) Changes in synaptic weight for active (beige) and inactive (blue) synapses based on voltage-dependent calcium levels in the dendritic shaft at the location of the inhibitory synapse. The dashed lines show the minimum/lower () and maximum/higher () thresholds. (Bottom) Functions for updating the thresholds of the upper panel, depending on the voltage-dependent calcium level. The asterisk denotes the calcium level where the curves for active and inactive synapses meet, which is a point of zero update.
Tables
Table 1
Excitatory plasticity parameters.
| Parameter | Description | Value |
|---|---|---|
| Learning rate for [Ca]NMDA-dependent plasticity | 1.5×10−5 µS mM−1 ms−1 | |
| Midpoint of LTP kernel (initial value) | 0.02 mM | |
| Controls width of LTP kernel | 1.0×103 mM−1 | |
| Learning rate for [Ca]L-type-dependent plasticity | 3×10–3 ms−1 mM−1 | |
| Constant LTD threshold | 7×10–5 mM | |
| Controls steepness of LTD threshold | 1.0×10 5 mM−1 | |
| Rate for shifting 𝜃ltp during dopamine peaks | 1×10–7 mM2 ms–1 | |
| Rate for shifting 𝜃ltp during dopamine pauses | 4×10–7 mM2 ms−1 | |
| Controls width of metaplasticity kernel | 334 mM–1 |
Table 2
Inhibitory plasticity parameters.
| Parameter | Description | Value |
|---|---|---|
| Controls steepness of the curves | 2.5×103 mM–1 | |
| Learning rate for active/inactive inhibitory weight | active: −0.055 , inactive: 0.055 µS-1 ms–1 | |
| Controls inhibitory weight | −1 | |
| Controls inhibitory weight | 3 | |
| Maximum synaptic strength | 0.005 µS | |
| Learning rate for modifying | 9×10-4 mM ms–1 | |
| Controls | −1 | |
| Controls | 3 | |
| Learning rate for modifying | –5×10–5 mM ms–1 | |
| Controls | −2 | |
| Controls | 3 | |
| Calcium concentration offset | 6×10–4 mM |
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Local, calcium- and reward-based synaptic learning rule that enhances dendritic nonlinearities can solve the nonlinear feature binding problem
eLife 13:RP97274.
https://doi.org/10.7554/eLife.97274.4
