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 
. It is believed that the Ca
influx through NMDA receptors is responsible for this potentiation 
and that the abrupt onset arises because the NMDA channel be in an open but blocked configuration before subsequent depolarization removes the Mg
. To assure a large Ca
influx and subsequent potentiation, it seems reasonable to assume that the depolarization that removes the Mg
block should occur near the peak of the NMDA activation. The Mg
removal by postsynaptic depolarization is extremely rapid 
but the NMDA activation has a finite rise time, so the peak of NMDA activation occurs a few milliseconds after the arrival of the presynaptic spike 
. Therefore, it seems likely that the maximum potentiation should occur when the presynaptic spike precedes the postsynaptic action potential by several milliseconds, and that depression could result from timing differences shorter than this. Thus, the biophysics of the NMDA receptor appears to support the idea of a temporal shift in the STDP window. The shape of the STDP window has been inferred from models of NMDA receptor kinetics and back-propagating action potentials 
. However, the millisecond timing of the transition from depression to potentiation was not investigated systematically, because its significance was not evident at that time. Nevertheless, in some parameterizations of such models a small depression domain for short positive pairing intervals has been reported 
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 transmission delay of the EPSP to the soma and that of the backpropagating action potential subtract from the delay we need for shifted STDP. For distal synapses where these delays are longer, there may be a higher probability that the causal bump falls out of the depression domain caused by the shift. This might be a mechanism for counterbalancing the attenuation of the EPSPs arising from distal dendrites 
along with other proposed mechanisms 
. It may explain the enhancement of LTD reported in studies of STDP at distal sites 
. If the delay becomes longer than the shift for very distal synapses, other mechanisms such as limits on synaptic strength must serve to stabilize STDP. Finally, if the speed of backpropagating action potential can be increased through modification of voltage-dependent conductances, the model predicts that synapses should be more readily depressed.
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 
and from paired recordings 
. However, the observed distribution of peak EPSP amplitudes has a heavier tail than the gamma distribution obtained from shifted STDP (see also 
). STDP is unlikely to be the only mechanisms involved in shaping the distribution of synaptic strengths. Nevertheless, shows that in presence of correlated input, this distribution can be quite broad. Thus, in the context of shifted STDP, a heavy-tailed distribution may be a sign of multiple correlated subgroups of input spike trains.
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 
. We also showed that a shifted version of the triplet model is stable and implements both Hebbian and anti-Hebbian competitions, as in the shifted pair-based model. It is worth noting that the suppression model 
is inherently stable without any shift and shows solely anti-Hebbian competition between correlated and uncorrelated inputs 
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.