# A criterion for the convergence of learning with spike timing dependent
plasticity

**R. Legenstein and W. Maass**

### Abstract:

We investigate under what conditions a neuron can learn by experimentally
supported rules for spike timing dependent plasticity (STDP) to predict the
arrival times of strong ``teacher inputs'' to the same neuron. It turns out
that in contrast to the famous Perceptron Convergence Theorem, which predicts
convergence of the perceptron learning rule for a strongly simplified neuron
model whenever a stable solution exists, no equally strong convergence
guarantee can be given for spiking neurons with STDP. But we derive a
criterion on the statistical dependency structure of input spike trains which
characterizes exactly when learning with STDP will converge on average for a
simple model of a spiking neuron. This criterion is reminiscent of the linear
separability criterion of the Perceptron Convergence Theorem, but it applies
here to the rows of a correlation matrix related to the spike inputs. In
addition we show through computer simulations for more realistic neuron
models that the resulting analytically predicted positive learning results
not only hold for the common interpretation of STDP where STDP changes the
weights of synapses, but also for a more realistic interpretation suggested
by experimental data where STDP modulates the initial release probability of
dynamic synapses.

**Reference:** R. Legenstein and W. Maass.
A criterion for the convergence of learning with spike timing dependent
plasticity.
In Y. Weiss, B. Schoelkopf, and J. Platt, editors, *Advances in Neural
Information Processing Systems*, volume 18, pages 763-770. MIT Press, 2006.