Intrinsic stability of temporally shifted spike-timing dependent plasticity

Abstract

Spike-timing dependent plasticity (STDP), a widespread synaptic modification mechanism, is sensitive to correlations between presynaptic spike trains and it generates competition among synapses. However, STDP has an inherent instability because strong synapses are more likely to be strengthened than weak ones, causing them to grow in strength until some biophysical limit is reached. Through simulations and analytic calculations, we show that a small temporal shift in the STDP window that causes synchronous, or nearly synchronous, pre- and postsynaptic action potentials to induce long-term depression can stabilize synaptic strengths. Shifted STDP also stabilizes the postsynaptic firing rate and can implement both Hebbian and anti-Hebbian forms of competitive synaptic plasticity. Interestingly, the overall level of inhibition determines whether plasticity is Hebbian or anti-Hebbian. Even a random symmetric jitter of a few milliseconds in the STDP window can stabilize synaptic strengths while retaining these features. The same results hold for a shifted version of the more recent "triplet" model of STDP. Our results indicate that the detailed shape of the STDP window function near the transition from depression to potentiation is of the utmost importance in determining the consequences of STDP, suggesting that this region warrants further experimental study.

Figure 1. Comparison of unshifted and shifted STDP.

A. The U-shaped steady-state distribution of synaptic strengths for conventional unshifted STDP. B. The unimodal steady-state distribution of synaptic strengths for shifted STDP (). C. The probability density of pairing intervals for presynaptic and postsynaptic spike trains. The blue area is the symmetric acausal contribution, and the pink area is the additional causal bump arising from postsynaptic spikes induced by the presynaptic input. D. Same as C, but for a stronger synapse. The causal bump is larger and closer to . E. The causal bump superimposed on the unshifted STDP window. The potentiation part of the STDP curve is red and the depression part blue. The causal bump falls entirely within the potentiation domain (red shading). F. Same as E, but for a stronger synapse. The causal bump still falls within the potentiation region. G. Same as E, but for shifted STDP. Part of the causal bump falls into the depression region (blue shading). H. Same as G, but for a stronger synapse. More of the causal bump falls into the depression region.

Figure 2. Shifted STDP stabilizes the distribution of synaptic strengths.

The horizontal axis is the value of the shift, the vertical axis is the synaptic strength and the gray level is the probability density of strengths, obtained by simulation. Solid line is the analytically calculated mean and dashed lines show the analytically calculated standard deviation around the mean. Insets show the distribution of synaptic strengths for different values of the shift. Solid curves are analytically calculated distributions. The arrows at the bottom of the horizontal axis of the main plot show the shift values corresponding to the insets.

Figure 3. The steady-state postsynaptic firing rate.

The steady state firing rate is plotted as a function of the input rates for excitation and inhibition. The inset shows the corresponding analytic result.

Figure 4. Synaptic competition through shifted STDP.

Cyan color indicates synapses with uncorrelated inputs, and magenta indicates correlated inputs. The rate of excitatory input is fixed at 10 Hz, and the correlation coefficient is 0.2 for correlated input spike trains. A. Steady-state distribution of synaptic strengths for an inhibitory rate of 10 Hz. Uncorrelated synapses become stronger than correlated. B. Steady-state distribution of synaptic strengths for an inhibitory rate of 20 Hz. Correlated synapses now become stronger than uncorrelated. C. Distributions of strengths for synapses receiving uncorrelated (top) and correlated (bottom) inputs as a function of the inhibitory input rate. The color level indicates the probability density of strengths. A transition from anti-Hebbian to Hebbian competition occurs at an inhibitory input rate of 14 Hz (dotted line). Arrows indicate the parameters for panels A and B. D. The transitional inhibitory rate as a function of correlation time constant. The transition takes place at lower inhibitory rates as the correlation time constant increases up to , then remains constant at for higher values. The insets show the full distribution of correlated and uncorrelated synaptic strengths as in C, for correlation time constants of , and .

Figure 5. The effect of the inhibitory input on synaptic competition.

A. Transition from mean-driven to fluctuation-driven firing regimes when the rate of the inhibitory input is increased. The black curve is the coefficient of variation of postsynaptic interspike intervals (), the blue curve is the mean free-running membrane potential in units of the spiking threshold, and the red curve is the standard deviation of the membrane potential in the same units. For inhibitory input rates greater than 14 Hz, there is an abrupt switch from the mean-driven to the fluctuation-driven regime, corresponding to the transition from anti-Hebbian to Hebbian competition (figure 4). B. Postsynaptic causal bumps due to uncorrelated (cyan) and correlated (magenta) input spikes for different mean synaptic strengths (shading) when the inhibitory input rate is 10 Hz. The blue area shows the depression domain and the red area is the potentiation domain. Note that the correlated causal bumps (magenta) fall almost entirely into the depression domain (blue shading) in this case, so the correlated synapses lose the competition. C. Same as panel b, but for an inhibitory input rate of 20 Hz. Note the heavy tail of the correlated causal bumps (magenta), which extend into the potentiation domain of the STDP window. These curves were obtained by numerical simulations, changing the mean of the steady-state distribution of correlated or uncorrelated synapses to the desired value for each curve. Because the correlated synapses arrive in unison, their causal bump is the aggregate effect of all of their spikes. To show the contribution of individual correlated spikes, comparable to that of the uncorrelated ones, we therefore normalized the magnitude of the causal bump of the correlated synapses by their average cluster size ().

Figure 6. Jittered STDP.

A. A random symmetric jitter of the unshifted STDP window (top) results in an effective window function (bottom) in which depression is dominant for short positive pairing intervals (blue shading). B. Jittered STDP stabilizes the distribution of synaptic weights. The horizontal axis is the standard deviation of the jitter (), the vertical axis is synaptic strength and the gray level indicates the probability density of strengths. For jitters smaller than 2 ms the distribution is bimodal, but for larger jitters it is stable and unimodal. C. The steady-state firing rate of the postsynaptic neuron as a function of the excitatory and inhibitory input rates when the jitter is 3 ms. D. Jittered STDP () implements both Hebbian and anti-Hebbian competition. As in figure 4, the top panel shows the distribution of uncorrelated synapses (cyan) and the bottom panel shows the distribution of correlated synapses (magenta), both as functions of the inhibitory input rate. The transition from anti-Hebbian to Hebbian competition occurs when the inhibitory input rate is about 50 Hz in this case.

Figure 7. The shifted triplet model.

A. The final distribution of weights for different values of maximum triplet potentiation () and depression (). Except for very high depression values, the distribution is unimodal and stable. We used the representative value of for both and (red dotted box) for the remaining results in this figure. B. The shift stabilizes the distribution of synaptic weights. The horizontal axis is the value of the shift, the vertical axis is the synaptic strength, and the gray level is the probability density of the strengths (as in figure 2), obtained by simulation. C. The steady-state firing rate of the postsynaptic neuron as a function of the excitatory and inhibitory input rates. D. The shift in the triplet model can implement both Hebbian and anti-Hebbian competition. As in figure 4, the top panel shows the distribution of the uncorrelated synapses (cyan) and the bottom panel shows the distribution of the correlated ones (magenta), as a function of the inhibitory input rate. The transition from anti-Hebbian to Hebbian competition occurs at an inhibitory input rate of 16 Hz.