Table 1.
Neuronal, synaptic, and plasticity parameters.
Fig 1.
A. Top-left: The STDP window with A+ < A−. Top-right: a triplet of spikes composed of two pre-post pairs with intervals Δt1 and Δt2. Bottom: the amount of synaptic modification in response to triplets, which is symmetric in the pair-based model. B. The average drift induced by the pair-based model on a population of excitatory synapses converging onto a single postsynaptic neuron, when A+ < A−. The black curve is a numerical evaluation of Eq 2 and the gray area is the simulation results. The half-width of the gray area is the standard error. The filled circle is the stable fixed point. The inset shows the w-dependent drift (Eq 3) C. The steady state distribution of synaptic weights obtained by simulation when A+ < A−. D. The steady state distribution of weights when half of the synapses receive correlated input (magenta) and the other half receive uncorrelated input (cyan). When A+ < A− correlated synapses are strengthened. E-H. The same as A-D, but for A+ > A−. Note that there is no stable fixed point in F, and that all the synapses are pushed to the upper bound in G and H. For these simulations, the constants of the STDP model were τ+ = τ− = 20 ms, A+ = 0.005 mV and A− = 1.0.1 A+ in A-D and A− = 0.005 mV and A+ = 1.0.1 A− in E-H.
Fig 2.
A. Schematic illustration of spike interactions in the triplet model in which previous presynaptic spikes induce extra depression (top) and previous postsynaptic spikes induce extra potentiation (bottom). B. Plasticity due to triplets of spikes: pre-post-pre triplets induce depression or weak potentiation (top left), and post-pre-post ordering induces mostly potentiation (bottom right). This figure is based on parameters fit to hippocampal data (Table 2).
Table 2.
Original parameters of the multi-spike STDP models used to generate Figs 2B, 4B, 7B and 9.
Fig 3.
Stability and competition in the triplet model.
A-E. Pair-based depression is larger than pair-based potentiation. A. Fixed points of ⟨w⟩ as functions of the ratio between postsynaptic potentiation and presynaptic depression parameters (Apost/Apre). When Apost/Apre is small, two nontrivial fixed points exist, one stable and one unstable. At higher values, they collide and disappear. When a stable fixed point exists (solid curve), the model is potentially competitive (dark gray area). B. Average drift of the weights, when Apost/Apre = 0.2. The gray area shows simulation results, and the solid curve is obtained from Eq (4). The filled circle depicts the stable fixed point and the open circle the unstable fixed point. The inset shows the w-dependent drift near the stable fixed point. C. Average drift of the weights when Apost/Apre = 1.2. The average weight has no nontrivial fixed points. D. Distribution of synaptic weights obtained from simulation. With parameters as in B and an initial mean of 0.4 mV, the final distribution is U-shaped (left). With an initial mean of 1.6, the final distribution clusters around the upper bound (right). Using parameters as in C, the final distribution also clusters around the upper bound (bottom). E. Synaptic competition for the parameters and initial values used in corresponding panels of D. Hebbian competition occurs only when the mean weight is stable and its initial value is below the unstable fixed point (left). F-J. Same as A-E, but when pair-based potentiation is larger than pair-based depression. F. The nontrivial fixed points disappear at lower values of Apost/Apre than in A, making the potentially competitive region smaller than in A (dark gray area). G-J. The same as B-E, but with pair-based potentiation larger than pair-based depression. Because the stable and unstable fixed points are close (G), competition does not occur even in the presence of a stable fixed point for the mean weight (J, left). For this figure, the time constants of presynaptic depression and postsynaptic potentiation were τpre = τpost = 40 ms, and the pair-based parameters of the model were the same as the pair-based model in Fig 1.
Fig 4.
A. Schematic illustration of spike interactions in the suppression model, in which the effect of the presynaptic spike in a pair is suppressed by a previous presynaptic spike (top), and the effect of the postsynaptic spike is suppressed by a previous postsynaptic spike (bottom). B. Plasticity in the suppression model induced by triplets of spikes: pre-post-pre triplets induce potentiation (top left), and post-pre-post triplets induce depression (bottom right).
Fig 5.
Stability and competition in the suppression model.
A. Fixed points of ⟨w⟩ as functions of the ratio between the potentiation and depression time constants. The stable fixed point disappears beyond the critical value τ+/τ− < 1.2. When the ratio approaches the critical value, the fixed point grows rapidly (gray area), leading to a stable distribution. B. The average drift when τ+/τ− = 1. The solid curve shows the analytical result (Eq 6) and the boundaries of gray shading is obtained by simulations. The filled circle is the stable fixed point. C. The average drift when τ+/τ− = 1.1. The stable fixed point moves to larger values than in B. D. The average drift when τ+/τ− = 1.5. No nontrivial fixed point exists. E. The partially stable bimodal steady-state distribution of weights corresponding to the parameters of B. F. The stable steady-state distribution of weights corresponding to the parameters of C. G. The unstable steady-state distribution of weights clustered around the upper bound corresponding to the parameters of D, when no stable fixed point exists. H-J. Competition between correlated and uncorrelated synapses with parameter corresponding to E-G. The competition is anti-Hebbian in all cases.
Fig 6.
Response of a neuron to a pair of presynaptic spikes and its consequences in the suppression model.
A. When the synapse is weak, the probability of a postsynaptic spike does not increase significantly from the baseline. The interval between postsynaptic spikes and also the pairing interval between the second presynaptic and the first postsynaptic spike are likely to be long. The result is a weak depression and also a weak suppression of potentiation. B. When the synapse is strong, the neuron is likely to fire in response to both presynaptic spikes, which results in strong depression and also strong suppression of potentiation.
Fig 7.
A. Schematic illustration of spike interactions in the NMDAR-based model. The presynaptic spike up-regulates frest, activates Mdn and depresses the synapse. The postsynaptic spike down-regulates frest, activates Mup and potentiates the synapse. B. Plasticity in the NMDAR-based model due to triplets of spikes with parameters as in Table 2. The effect is asymmetric, with pre-post-pre triplets inducing potentiation (top left) and post-pre-post depression (bottom right).
Fig 8.
Stability and competition in the NMDAR-based model.
A. Fixed points of ⟨w⟩ as functions of the ratio between the maximum potentiation and depression parameters. When A+/A− is smaller than 0.042, two nontrivial fixed points exist. At higher values, they collide and disappear. B. The w-dependent drift at the stable fixed point, as a function of A+/A−, which changes sign at A+/A− = 0.025. In the dark gray region, a stable fixed point exists and the w-dependent drift is negative. In the light gray region a stable fixed point exists and the w-dependent drift is positive, and in the white region there is no stable fixed point. C-E. The average drift when A+/A− is 0.01, 0.03 and 0.05, respectively. Filled circles represent stable fixed points and the open circle an unstable fixed point. The gray shading is the result of simulations, and the solid curve is the analytical result. F-H. The steady-state distributions corresponding to the parameters in C-E. I-K. Synaptic competition between correlated and uncorrelated synapses corresponding to the parameters in C-E. Parameters are , and the time constants are as in Table 2.
Fig 9.
Stability and competition in multi-spike STDP models with soft bounds.
A. Steady-state distribution of synaptic strengths in the triplet model with soft bounds. B. Steady-state distribution of synaptic strengths in the suppression model with soft bounds. C. Steady-state distribution of synaptic strengths in the NMDAR-based model with soft bounds. Insets: steady-state distribution of weights when half of the synapses receive correlated input (magenta) and the other half receive uncorrelated input (cyan). In each case, the original parameters (Table 2) are used.
Table 3.
Summary of stability/plasticity in STDP models.
Fig 10.
Relationships between the three multi-spike STDP models.
If the second messengers activate instantaneously, the NMDAR-based model is qualitatively equivalent to the suppression model (top, right). If there exists an infinite reservoir of resting NMDARs and inactive second messengers, the NMDAR-base model reduces to the triplet model (bottom-left). If both assumptions are fulfilled, the NMDAR-based model reduces to the pair-based model (bottom- right).