2.3.1. Model
The stop learning model of long-term plasticity has been introduced in Brader et al. (2007), based on earlier work in Fusi et al. (2000). The model represents a synapse with two stable states, potentiated and depressed, whereby the state transition between both stable states is regulated via a continuous internal state X(t) of the synapse. X(t) is influenced by a combination of pre- and postsynaptic activity, namely the presynaptic spike time tpre and the value of the neuron membrane voltage Vmem(t). A presynaptic spike arriving at tpre reads the instantaneous values Vmem(tpre) and C(tpre). The conditions for a change in X depend on these instantaneous values in the following way:
(5)  X → X + a   i f   { V m e m   ( t p r e   ) > θ V    a n d                                                     θ u p  l  < C ( t p r e   ) < θ u p  h  }      (6)  X → X − b   i f   { V m e m   ( t p r e   ) ≤ θ V    a n d                                                     θ d o w n  l  < C ( t p r e   ) < θ d o w n  h  } ,      where a and b are jump sizes and θV is a voltage threshold. In other words, X(t) is increased if Vmem(t) is elevated (above θV) when the presynaptic spike arrives and decreased when Vmem(t) is low at time tpre. The θlup, θhup, θldown, and θhdown are thresholds on the calcium variable. The calcium variable C(t) is an auxiliary variable (see Brader et al., 2007 for details) that provides a low-pass filter of the postsynaptic spikes. This gives the ability to stop the learning based on thresholded, long-term averages of postsynaptic activity. In the absence of a presynaptic spike or if stop learning is active [i.e., C(t) hits the respective threshold], then X(t) drifts toward one of two stable values:
(7)  d X  d t   = α        i f   X > θ X        (8)  d X  d t   = − β   i f   X ≤ θ X         The bistable state of the synapse is determined according to whether X(t) lies above or below the threshold θX. Computationally, this model is interesting because through X(t) it can learn a graded response to an input pattern even though the output weight of the synapses is binary. The model also has some biological veracity, being sensitive to pre-post and post-pre spike patterns in a manner similar to the well-known spike time dependent plasticity (Brader et al., 2007).