A triplet spike-timing-dependent plasticity model generalizes the Bienenstock-Cooper-Munro rule to higher-order spatiotemporal correlations

Abstract

Synaptic strength depresses for low and potentiates for high activation of the postsynaptic neuron. This feature is a key property of the Bienenstock-Cooper-Munro (BCM) synaptic learning rule, which has been shown to maximize the selectivity of the postsynaptic neuron, and thereby offers a possible explanation for experience-dependent cortical plasticity such as orientation selectivity. However, the BCM framework is rate-based and a significant amount of recent work has shown that synaptic plasticity also depends on the precise timing of presynaptic and postsynaptic spikes. Here we consider a triplet model of spike-timing-dependent plasticity (STDP) that depends on the interactions of three precisely timed spikes. Triplet STDP has been shown to describe plasticity experiments that the classical STDP rule, based on pairs of spikes, has failed to capture. In the case of rate-based patterns, we show a tight correspondence between the triplet STDP rule and the BCM rule. We analytically demonstrate the selectivity property of the triplet STDP rule for orthogonal inputs and perform numerical simulations for nonorthogonal inputs. Moreover, in contrast to BCM, we show that triplet STDP can also induce selectivity for input patterns consisting of higher-order spatiotemporal correlations, which exist in natural stimuli and have been measured in the brain. We show that this sensitivity to higher-order correlations can be used to develop direction and speed selectivity.

Fig. 1.

The triplet STDP rule. (A) Synaptic depression is induced as in classical pair-based STDP using spike pairs separated by Δt₁ = t^post − t^pre < 0. Synaptic potentiation is induced using triplets of spikes consisting of two postsynaptic spikes and one presynaptic spike on the basis of the timing interval between them Δt₁ = t^post − t^pre > 0 and Δt₂ = t^post − t′^post > 0. (B) Synaptic change as a function of the time between pre- and postsynaptic spikes in a protocol where 60 pairs were presented at different frequencies ρ = 0.1, 20, and 50 Hz. Depression predominated at low frequency, whereas potentiation was more prevalent at high frequencies. The data points are experiments are from ref. and the lines were generated with the triplet STDP rule with the parameters taken from ref. .

Fig. 2.

Triplet STDP induces selectivity with rate-based patterns. (A) Evolution of the weights (Right) for 10 rate-based patterns (uniformly spaced Gaussian profiles across the 100 inputs) determining the inputs’ firing rates (Left) presented to a feedforward network. The selected pattern corresponds to a Gaussian profile with r_min/r_max = 5/55 and σ = 15 Hz. (B) Mean (±SEM) selectivity (for 10 trials) as a function of the Gaussians’ SD σ and for Gaussian profiles with different ratios of background to peak firing rates r_min/r_max = {0/55, 5/55, 10/55} (solid lines, the triplet STDP rule; dashed lines, the BCM rule). The Gaussian profiles below illustrate the amount of overlap for two neighboring Gaussians. Numerical simulations implementing the differential form of the triplet STDP were performed in A and B. (C) Weight change Δw as a function of postsynaptic activity for three different input firing rates, which determine the threshold θ for weight modification. Symbols denote numerics and lines analytics. (D) 2D phase plane analysis for the analytically derived weight equation with orthogonal rate-based patterns. Nullclines in green and purple intersect at the equilibria shown in red. (E) An example trajectory for the two weights attracted to one of the stable nodes in D. (a.u., arbitrary units).

Fig. 3.

Triplet STDP induces selectivity with correlation-based patterns. (A) Ten correlation-based patterns that have the same firing rates, but different correlation strength. (B) Evolution of the weights illustrates selectivity in the case of 10 correlation-based patterns. The firing rate of each of the 100 inputs was set to 10 Hz: 90 inputs had no correlations and 10 neighboring inputs (one of 1–10, 11–20, … , 91–100 for each pattern) had strong spatial correlations (90% identical spikes). (C) Same as B except for the 10 correlated inputs in each pattern, for which exponentially decaying correlations with a time constant of 5 ms were used. Numerical simulations implementing the differential form of the triplet STDP were performed in B and C. (D) The average weight change Δw (for 100 weights) was computed for different initial conditions w₀ after 100 s. The symbols denote numerical results obtained by simulating the differential form of the triplet rule, and the lines indicate a semianalytic solution by numerically solving Eq. 6 given an initial condition w₀ for 100 s. The average weight change was plotted as a function of the postsynaptic firing rate given by ν = w₀ρ, where ρ was the input firing rate. Here we simulated two networks where the inputs had the same firing rate (10 Hz) and exponentially decaying correlations with a timescale of 10 ms. The correlation peak for the curve in black (SI Text) was half of the correlation peak for the curve in red (γ = 9.09, λ = 9.09); see Inset. (E) 2D (two groups of inputs) phase plane analysis for correlation-based patterns. Nullclines in green and purple intersect at the unstable fixed points shown in red. Imposing a lower bound at 0 resulted in stable maximally selective fixed points on the axes shown in black. (F) An example trajectory for the two weights attracted to one of the black equilibria in E. (G) Percentage of cases (over 100 trials) where all of the synapses from 1 of 10 input patterns (each consisting of 10 inputs) potentiate (Eq. 6). Same scenario as C, but for different correlation time constants τ_c. Correlations were symmetric and exponentially distributed (Inset).

Fig. 4.

Triplet STDP, and not pair-based STDP, can distinguish between patterns determined by third-order correlations. (A) Two patterns were randomly presented to a feedforward network: The inputs in the two patterns had the same firing rates and the same pairwise correlations. The patterns differed only by the presence or absence of third-order correlations in half of the inputs (illustrated with the red triplets of spikes and the colored background). The probability of presenting pattern 1 was varied, e.g. of 10 pattern presentations, pattern 1 was presented with probability 0.8. (B) The evolution of the weights under the triplet STDP rule demonstrating an example where pattern 1 (inputs 1–3) wins. (C) Pattern 1 was presented to the network at different probabilities and the mean ± SEM of the probability that pattern 1 wins was computed for 200 simulations runs: triplet STDP (black symbols) and pair STDP (red symbols). (D) The triplet STDP rule is sensitive up to third-order correlations, but not to higher-order correlations. Pattern 1 was always presented and consisted of two groups of five neurons each. Both groups had the same correlations up to (but not including) order k (horizontal axis). Group 1 had nonzero ≥k^th-order correlations, and group 2 had zero ≥k^th-order correlations. The mean ± SEM of the probability that pattern 1 wins (i.e., all of the weights of group 1 potentiate) was computed for 200 simulation runs. Dashed lines correspond to chance level.

Fig. 5.

Triplet STDP leads to spatiotemporal receptive field development. (A) Four different bars (horizontal, vertical, and the two diagonals on a 9 × 9-pixels image) were presented as inputs to a feedforward network with a single postsynaptic neuron; each bar can move in one of two directions, giving a total of eight patterns. (B) Time evolution of the 81 synaptic weights. (C) Final weights reordered in a grid corresponding to the input location. (D) Histogram of the postsynaptic firing rate plotted after convergence of the weights, at the end of the learning in B. The firing rates shown resulted from the presentation of the eight different patterns (four orientations and two directions) averaged over 200 s. (E) After learning with different training presentation times (5, 7, 10, 12, 17, 20, 22, and 25 ms), the weights were frozen during a testing phase. The pattern (of the eight patterns) that resulted in the highest firing rate at the training presentation time was presented again to the network at different tested presentation times (5, 7, 10, 12, 17, 20, 22, and 25 ms) while the firing rate was measured. The best tested presentation time for which the firing rate (averaged over 100 s) was the highest is plotted against the training presentation time (mean ± SEM over 10 trials).