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Conceived and designed the experiments: BB LFA. Performed the experiments: BB LFA. Analyzed the data: BB LFA. Contributed reagents/materials/analysis tools: BB LFA. Wrote the paper: BB LFA.

The authors have declared that no competing interests exist.

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.

Synaptic plasticity is believed to be a fundamental mechanism of learning and memory. In spike-timing dependent synaptic plasticity (STDP), the temporal order of pre- and postsynaptic spiking across a synapse determines whether it is strengthened or weakened. STDP can induce competition between the different inputs synapsing onto a neuron, which is crucial for the formation of functional neuronal circuits. However, strong synaptic competition is often incompatible with inherent synaptic stability. Synaptic modification by STDP is controlled by a so-called temporal window function that determines how synaptic modification depends on spike timing. We show that a small shift, or random jitter, in the conventional temporal window function used for STDP that is compatible with the underlying molecular kinetics of STDP, can both stabilize synapses and maintain competition. The outcome of the competition is determined by the level of inhibitory input to the postsynaptic neuron. We conclude that the detailed shape of the temporal window function is critical in determining the functional consequences of STDP and thus deserves further experimental study.

Hebbian synaptic plasticity can effectively organize neural circuits in functionally useful ways, but only when implemented in a manner that induces competition among synapses

Synaptic competition and synaptic stability (meaning that synapses reach a stable equilibrium distribution independent of bounds on their strengths) are desirable but conflicting features of Hebbian synaptic plasticity. For example, the instability of STDP mentioned in the previous paragraph can be eliminated by introducing strength-dependent modification

To study the effects of STDP on synaptic strengths, we simulated a single spiking neuron that receives excitatory and inhibitory presynaptic spike trains with Poisson statistics at rates

Parameter | Symbol | Default value |

Membrane time constant | ||

Spiking threshold | ||

Resting membrane potential | ||

Maximum potentiation amplitude | ||

Maximum depression amplitude | ||

Potentiation time constant | ||

Depression time constant | ||

Window shift | ||

Synaptic time constant | ||

Number of excitatory synapses | ||

Number of inhibitory synapses | ||

Inhibitory synaptic strength | ||

Excitatory input rate | ||

Inhibitory input rate |

With conventional, unshifted STDP (

The total effect of a sequence of pre- and postsynaptic action potentials on the strength of a synapse can be computed by multiplying the STDP window function by the probability of a spike pair appearing with time difference

When there is no shift in the STDP window, the causal bump falls entirely within the potentiation domain (

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.

For a more quantitative evaluation of shifted STDP, we computed the steady-state solution of the Fokker-Planck equation governing the distribution of synaptic strengths

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.

STDP has an interesting regulatory effect on the steady-state firing rate of a neuron

The stabilization of synaptic strengths discussed in the previous section arises from the change of size and shape of the causal bump seen in

In general, we expect the firing rate of a neuron to increase when its excitatory inputs fire more rapidly, and this is exactly what occurs for excitatory input rates below about 10 Hz in

Shifted STDP also has a buffering property on changes in the inhibitory input rate. In presence of strong inhibitory input, the postsynaptic firing rate falls. This broadens the acausal part of the spike-pair distribution, lowering the chance for pairs to fall into the depression domain caused by the shift and, thus, resulting in more potentiation. However, in this case, the effect is not strong enough to overcome the expected tendency of the postsynaptic rate to be suppressed by inhibition (

Hebbian plasticity in general and STDP in particular allows neurons to become selective to correlated subsets of their inputs, but this requires synaptic competition

Interestingly, with shifted STDP the outcome of the competition depends on the rate of inhibitory input to the neuron. When the rate of inhibitory input is 10 Hz for the parameters we use, the synapses receiving correlated spikes end up weaker than the synapses receiving uncorrelated spikes (

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.

These results were obtained using spike trains with zero time-lag correlations, meaning that for any two correlated spike trains, a subset of spikes is perfectly synchronous. More realistic spike correlations can be generated by including a small random jitter in the timing of the synchronous spikes. The mean of this jitter determines the correlation time constant. Breaking perfect synchrony does not change the above results qualitatively. However, the rate of inhibitory input needed to transition from anti-Hebbian to Hebbian competition is sensitive to the correlation time constant (

The dependence of the outcome of synaptic competition on the level of inhibitory input can be explained by evaluating the effect of inhibition on the firing regime of the postsynaptic neuron. When the inhibitory input to a neuron is low, it operates in a “mean-driven” regime, meaning that the time-averaged “free-running” membrane potential (that is, the membrane potential if the spike generation mechanism is turned off) is above the firing threshold

The model neuron we study traverses these regimes as the firing rate of its inhibitory inputs is varied (

Recall that the causal bump is the excess probability of postsynaptic firing caused by an incoming input spike. As mentioned previously, the effect of shifted STDP on the distribution of synaptic strengths can be explained by considering the shape of the postsynaptic causal bump in relation to the STDP temporal window. When the postsynaptic neuron is in the mean-driven regime, the membrane potential rises rapidly to the threshold. As a result, presynaptic action potentials can only enhance postsynaptic firing if they occur during a relatively short time-interval prior to the postsynaptic spike. This means that the causal bump decays rapidly for longer intervals. The causal bump also has a higher amplitude and decays more rapidly for stronger synapses (

When the postsynaptic neuron fires in the fluctuation-driven regime, the membrane potential spends a considerable time near but below the firing threshold before spiking. As a result, presynaptic input can affect postsynaptic firing over a longer time interval than in the mean-driven regime. This makes the causal bump broader than in the mean-driven case (

The transition from the mean-driven to the fluctuation-driven regime and correspondingly from anti-Hebbian to Hebbian competition is quite abrupt. This may be due to the interplay between the correlated inputs and the firing mode of the neuron. Correlated inputs increase membrane potential fluctuations and spiking irregularity

It is not necessary to introduce an explicit shift into the STDP window to assure stability. Any mechanism that causes depression to dominate over potentiation for short positive pairing intervals will have the same qualitative effect. One such mechanism is a symmetric random jitter introduced into an unshifted STDP window that has

Simulations show that jittered STDP has all the qualitative properties of shifted STDP, although the maximum depression must be set to be greater rather than the maximum potentiation (we take

A pair-based STDP model cannot account for all experimentally observed spike-timing dependent synaptic modifications. When bursts of spikes are induced in the pre- and postsynaptic neurons, frequency dependence is observed for both pre-after-post and post-after-pre pairings. The magnitude of LTP, but not LTD, increases with burst frequency and, at high burst frequency, LTP is induced regardless of the ordering of the pre- and postsynaptic spikes

In the triplet model, a “2 pre/1 post” ensemble of spikes exerts an extra depression (the triplet depression) in addition to the usual pre-post pairing effect. The triplet depression has its maximum value

Simulations show that the final distribution of weights is stable and unimodal (

We have shown that a slight shift in the effective STDP temporal window, such that postsynaptic spikes occurring shortly after presynaptic action potentials cause synaptic depression, can stabilize the distribution of synaptic strengths without loss of competition, both in pair-based and triplet-based models. The shift can be explicitly implemented in the STDP window or achieved by other means such as a symmetric spike-by-spike random jitter. In fact, any mechanism that causes synaptic depression for small but causal (positive by our convention) pre-post spike intervals should lead to the stabilization and other effects we report. What biophysical mechanisms could cause this to occur?

The sharp transition between depression and potentiation in STDP appears to be due to the abrupt onset of long-term potentiation

Typically in electrophysiological recordings, action potentials are measured at the soma, but what matters for STDP is the timing of the events at the synapse. More precisely, the timing of the postsynaptic EPSP and that of the backpropagating action potential to the synapse control plasticity. Transmission delays may have their own interesting computational properties. For example, it has been shown that STDP in the presence of axonal transmission delays can have a desynchronizing effect on population bursts and a synchronizing effect on random spiking in a recurrent network

The most direct test of the shifted STDP hypothesis would be to observe the effect of almost synchronous pre- and postsynaptic spikes on synaptic strength. However, the results of such experiments could be difficult to interpret because of confounding factors such as the physiological delays mentioned above. For example, if the pre- and postsynaptic spikes are induced exactly at the same time, the timing of their arrival at the synapse is not necessarily synchronous. If a shift in the STDP window function acts as a stabilizing mechanism, synapses should get depressed when postsynaptic spikes are generated by presynaptic spikes with short latency. Therefore, as an alternative experiment we suggest inducing spikes only in the presynaptic neuron and allowing the postsynaptic firing to be affected by this presynaptic activity. One possible way to perform such an experiment is to hold the voltage of the postsynaptic neuron close to its firing threshold, so that individual EPSPs can induce a postsynaptic spike. In this case, if there is a stabilizing shift in the STDP window, strong synapses that induce short-latency postsynaptic action potentials abruptly should get depressed.

Shifted STDP results in a unimodal distribution of synaptic strengths. This finding is in agreement with the measurements of quantal synaptic currents

The synapses in the model we considered were current-based, meaning that each excitatory or inhibitory input injects a current waveform to the neuron regardless of the value of its membrane potential. We have also studied an analogous model with conductance-based synapses, and this does not qualitatively change the reported results. These results show that the outcome of competition between correlated and uncorrelated spike trains with shifted STDP depends on the firing state of the postsynaptic neuron, which can be controlled by the rate of its inhibitory inputs. This allows for a dynamic switching between anti-Hebbian and Hebbian forms of plasticity, and it might be related to the role of local inhibitory interneurons in switching the activity-dependent development of visual cortical circuits during the critical period

In conclusion, a slightly shifted STDP window stabilizes synaptic strength, buffers firing rates, and can implement different modes of synaptic competition. The required shift may arise from properties of the NMDA receptor, or from random jitter. In light of their importance in determined the outcome of synaptic plasticity, we argue that the properties of STDP for short pairing intervals, which have not yet been clearly resolved, warrant a more detailed investigation.

The membrane potential of the integrate-and-fire model neuron obeys

Each presynaptic action potential at an excitatory or inhibitory synapse induces an abrupt jump into the corresponding synaptic input (

In all simulations, the synaptic strengths were initialized randomly from a uniform distribution over the range 1–5 mV. For each parameter regime, the simulations were run for

The triplet model

To study synaptic competition, half of the excitatory input spike trains were correlated. To generate Poisson spike trains with homogeneous pairwise (zero-lag) correlations, we used the method developed by Kuhn et al

The evolution of the distribution of synaptic strengths is described by the Fokker-Planck equation

The terms

When the synaptic strengths are changing due to STDP, the only relevant stochastic variable is the interval between the pre- and postsynaptic spike pairs. If a pairing of pre- and postsynaptic spikes occurs with interval

We approximate the spiking behavior of the integrate-and-fire neuron by that of a linear Poisson neuron firing at the same rate

If we substitute 16 into equation 9, we obtain

Finally, by inserting equations 17 into equation 8, we obtain the steady-state distribution

For the above distribution 19 to be normalizable,

If

We thank Steve Siegelbaum for helpful comments.