STDP Rule

Spike Timing Dependent Plasticity (STDP) is a variant of the Hebbian rule specifically for synapses between spiking neurons. In STDP, the relative timing of pre- and post-synaptic spikes determines whether the synapse is strengthened (long-term potentiation, LTP) or weakened (long-term depression, LTD).

When the pre-synaptic neuron spikes before the post-synaptic neuron, the synapse is strengthened (LTP). When the pre-synaptic neuron spikes after the post-synaptic neuron, the synapse is weakened (LTD).
The update depends on the spike timing difference \(u = t_{post} - t_{pre}\):

\[\Delta w = \eta W_{+/-} \exp \left( -\frac{|u|}{\tau_{+/-}} \right)\]

where:

  • η is the learning rate
  • W+/− are the learning rates for LTP and LTD
  • τ+/− are the time constants for LTP and LTD
  • The sign of \(\Delta w\) depends on the sign of the weight, so inhibitory synapses also increase in absolute value under the same conditions as excitatory ones.

This window can be reversed (anti-Hebbian behavior) by setting the Hebbian parameter to false, meaning anti-causal relationships produce LTP and causal ones produce LTD.

STDP only applies to spiking neurons. Using non-spiking neurons with this rule is not recommended and can harm performance. If either the pre- or post-synaptic neuron is not spiking, the synapse behaves like a clamped synapse.

Source: Jean-Philippe Thivierge and Paul Cisek (2008), Nonperiodic Synchronization in Heterogeneous Networks of Spiking Neurons, Journal of Neuroscience; and the Scholarpedia article on STDP.

Also on anti-hebbian STDP: https://journals.physiology.org/doi/pdf/10.1152/jn.00551.2006

Parameters

  • Tau plus: Time constant \(\tau_{+}\) for LTP (weight strengthening when pre fires before post). Smaller values narrow the window within which LTP is applied.
  • Tau minus: Time constant \(\tau_{-}\) for LTD (weight decay when post fires before pre). Smaller values narrow the window within which LTD is applied.
  • W+: Learning rate for LTP case. Controls the magnitude of LTP changes.
  • W-: Learning rate for LTD. Controls magnitude of LTD changes.
  • Learning rate: Global learning rate \(\eta\).
  • Hebbian: If true, use hebbian learning, else anti-hebbian