Stability and Competition in Multi-spike Models of Spike-Timing Dependent Plasticity
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.