# What can a Neuron Learn with Spike-Timing-Dependent Plasticity?

**R. Legenstein, C. Naeger, and W. Maass**

### Abstract:

Spiking neurons are very flexible computational modules, which can implement
with different values of their adjustable synaptic parameters an enormous
variety of different transformations F from input spike trains to output
spike trains. We examine in this letter the question to what extent a spiking
neuron with biologically realistic models for dynamic synapses can be taught
via spike-timing-dependent plasticity (STDP) to implement a given
transformation F. We consider a supervised learning paradigm where during
training, the output of the neuron is clamped to the target signal (teacher
forcing). The well-known perceptron convergence theorem asserts the
convergence of a simple supervised learning algorithm for drastically
simplified neuron models (McCulloch-Pitts neurons). We show that in contrast
to the perceptron convergence theorem, no theoretical guarantee can be given
for the convergence of STDP with teacher forcing that holds for arbitrary
input spike patterns. On the other hand, we prove that average case versions
of the perceptron convergence theorem hold for STDP in the case of
uncorrelated and correlated Poisson input spike trains and simple models for
spiking neurons. For a wide class of cross-correlation functions of the input
spike trains, the resulting necessary and sufficient condition can be
formulated in terms of linear separability, analogously as the well-known
condition of learnability by perceptrons. However, the linear separability
criterion has to be applied here to the columns of the correlation matrix of
the Poisson input. We demonstrate through extensive computer simulations that
the theoretically predicted convergence of STDP with teacher forcing also
holds for more realistic models for neurons, dynamic synapses, and more
general input distributions. In addition, we show through computer
simulations that these positive learning results hold not only 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, C. Naeger, and W. Maass.
What can a neuron learn with spike-timing-dependent plasticity?
*Neural Computation*, 17(11):2337-2382, 2005.