Figure 1.
Choosing Slow Directions of the Input
Finding the direction of least variance in the time derivative of the input (which is part of the SFA algorithm) can be replaced by finding the direction of maximum variance in an appropriately low-pass filtered version of the input signal.
Figure 2.
Filtered Hebbian Learning Rule
Input and output signals are filtered (downward arrows). The weight change is the result of applying the Hebbian learning rule on the filtered signals (square box and upward arrow). Thereby, the variance of the filtered version of the output is maximized without actually filtering the output during processing.
Figure 3.
Relation between the EPSP and the Learning Window
The power spectrum is the Fourier transform of the effective learning window W0, which in turn is the convolution of the learning window W and the EPSP ε. The figure shows the learning windows required for SFA for three different EPSP durations (τ = 4, 40, 400 ms). The maximal input frequency νmax was 1 / (40 ms) in all plots.
Figure 4.
Comparison of the Learning Window with Experimental Data
The plot compares the theoretically predicted learning window with experimental data from hippocampal pyramidal cells as published by Bi and Poo [16] (larger plot in the middle). Instead of the ideal power spectrum with the abrupt cutoff at νmax as stated in Equation 20, a Cauchy function with γ = 1 / 15 ms was used (top left, the dashed line is
for νmax = 1 / (40 ms)). Again, the EPSP decay time was τ = 40 ms. This learning window corresponds to an implementation of the “trace rule” [1,4,6] for a decay time of the exponential filter of 15 ms.