Fig. 1: Description of the predictive plasticity rule.
From: Sequence anticipation and spike-timing-dependent plasticity emerge from a predictive learning rule
a Illustration of the model and the computational graph corresponding to the learning algorithm. Top: at time step t, the neuron computes a prediction of the new input xt from the previous membrane potential vt−1 and synaptic weight vector wt−1 (see Equation (2)). The prediction error is used to drive synaptic plasticity and update the synaptic weight vector wt−1 (see Equation (3)). Bottom: the neuron updates its membrane potential by encoding the actual input xt via the learned weight vector wt and its previous internal state vt−1 (see Equation (1)). If the voltage exceeds the threshold, an output spike is emitted (shown in yellow) and this spiking event reduces the membrane potential by a constant value at the next time step. Otherwise, the value of the membrane potential vt is kept and passed to the next time step. b In the simulation illustrated here, we considered a pattern of two pre-synaptic spikes from two different pre-synaptic neurons with a relative delay of 4 ms. Shown are the dynamics of the membrane potential at the first training epoch and after 100 iterations. The neuron learns to fire ahead of the input that arrives at 6 ms (i.e. pre-syn neuron 2). c Top: Dynamics of the weights for different initial conditions (i.e. the weights at epoch 0). The unbroken and dashed lines correspond, respectively, to the pre-synaptic inputs arriving at 2 ms (w1, pre-synaptic neuron 1) and 6 ms (w2, pre-synaptic neuron 2). Bottom: evolution of the output spike times across epochs. The bottom and top plot have the same color code. d The flow field in the parameter space was obtained by computing the difference between the weight vector (w1, w2) in the first epoch and after 10 epochs. The blue lines represent the partition given by the number of spikes that are fired. Note that when the synaptic weights are larger, the neuron fires more spikes. The black arrow shows the trajectory of the weights obtained by training the model for 500 epochs with initial conditions w0=(0.005,0.005). The shaded region shows the section of the parameter space where the neuron fires ahead of the input at 6 ms from neuron 2.
