## Analysis of oscillatory weight changes from online learning with filtered spiking feedback

#### Tech Report, 2017

Aaron R. Voelker, Chris Eliasmith

### Abstract

Prescribed Error Sensitivity (PES) is a biologically plausible supervised learning rule that is frequently used with the Neural Engineering Framework (NEF). PES modifies the connection weights between populations of spiking neurons to minimize an error signal. Continuing the work of Voelker (2015), we solve for the dynamics of PES, while filtering the error with an arbitrary linear synapse model. For the most common case of a lowpass filter, the continuous-time weight changes are characterized by a second-order bandpass filter with frequency $\omega = \sqrt {\tau ^{-1} \kappa \|{\bf a}\|^2 }$ and bandwidth $Q = \sqrt {\tau \kappa \|{\bf a}\|^2 }$, where τ is the exponential time constant, κ is the learning rate, and ${\bf a}$ is the activity vector. Therefore, the error converges to zero, yet oscillates if and only if $\tau \kappa \|{\bf a}\|^2 > \frac {1}{4}$. This provides a heuristic for setting κ based on the synaptic τ, and a method for engineering remarkably accurate decaying oscillators using only a single spiking leaky integrate-and-fire neuron.

#### CTN Tech Report

Issn
CTN-TR-20171001-013