Stable and robust spike timing dependant plasticity

Im afraid there is more techie maths in this post :(. I recently posted here about spike timing dependant plasticity and how it can create unimodal or bimodal synaptic weight distributions depending on whether a term is added to the weight change function to allow dependence on the existing value of the synaptic weight.

Babadi et al show that by adding a delay (d) to the exponential term in additive spike timing depenant plasticity weight change equation (i.e. -(|Δt|-d) / τ) one can stabalize the distribution of synaptic weights to be unimodal instead of bimodal even when no limits are imposed. This is because apparently the relative strength of synapse induces a causal bump near Δt=0 invoking stronger increases the stronger the weight.This makes sense as the synaptic weight will cause the post-synaptic neuron to fire more quickly and often if it is stronger. This bump can be seen in the image below where on left side of the y axis is the depressive exponential term (though not shown with negative weighting) and on the other side is the potentiating exponential term with the bump:

The delay term makes the causal bump fall into the region where depression occurs.As in the image below (this time the depressive exponential part is shown with its negative weighting):

As the synapse gets stronger, a larger portion falls into the depression area, both because the causal bump gets bigger and because it moves closer to Δt=0. This prevents further growth of the synaptic strength and therefore stops saturation around the bimodal MIN and MAX possible values for synaptic weights.

Different studies in synchrony (for a description of synchrony see this post) and learning have shown it irrelevant whether excitatory post synaptic potential arrive shortly before or after the post-synaptic spike, but long term depression of synapses occurred when the same synchronous group input was oscillated 180 degrees out of phase so that the excitatory post-synaptic potentials arrived during the troughs of the oscillation, suggesting a certain robustness to the influence of synchronised activity on STDP (read this pdf for more info).

This finding raises interesting questions as to a possible interplay with a delay term introduced by Babadi et al.

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