Table 1.

Synaptic filtering: Examples of differential operators and their corresponding temporal filters

Q	s(t)
1	δ(t)
$\left(1+\frac{1}{\alpha }\frac{\mathrm{d}}{\mathrm{d}t}\right)$	$\alpha {\mathrm{e}}^{-\mathrm{\alpha t}}\mathrm{\Theta }\left(t\right)$
$\left(1+\frac{1}{{\alpha }_1}\frac{\mathrm{d}}{\mathrm{d}t}\right)\left(1+\frac{1}{{\alpha }_2}\frac{\mathrm{d}}{\mathrm{d}t}\right)$	${\left(\frac{1}{{\alpha }_1}-\frac{1}{{\alpha }_2}\right)}^{-1}\left[{\alpha }_1{\mathrm{e}}^{-{\alpha }_{1}t}-{\alpha }_2{\mathrm{e}}^{-{\alpha }_{2}t}\right]\mathrm{\Theta }\left(t\right)$

The first example shows pulsatile coupling, where no synaptic filtering has been applied to the incoming spike. The second type of filter shown is an exponentially decaying function, which accounts for the slow decay of the instantaneous pulse. It does not however account for the time it take a synapse to process the incoming action potential, increasing instantaneously to the maximum value as soon as the spike arrives. The last example takes into account this synaptic processing delay, increasing smoothly to its peak value and then decaying exponentially back to zero. Note that Θ(t) represents the Heaviside function