Notes

Sequence-To-Sequence Learning Because zero error was achieved with simple tasks using different types of feedback signal, the learning rule can be put to work in harsher conditions, under more challenging tasks: making it associate a spatio-temporal target output pattern to a repeating frozen Poisson noise input. In this case, a larger, 3-layer net was used (100 in, variable number hidden, 100 out), but an output pattern matching the target was achieved with only 32 hidden neurons. The random feedback performs worse than a network that was trained without a hidden layer, but with symmetrical weights.

What about Unsupervised Learning? A Spiking Auto-Encoder with "Gaussian" Input

The rule can also be used as an auto-encoder network, since it can be provided the same pattern as both input and output. It is able to reconstruct the output pattern with high fidelity while having a number of hidden units smaller than the number of units in the input and output layers.

Spiking Nets and Temporal Coding The RNN term is used in its widest sense, that of networks with states evolving in time based on well/defined dynamic recurrent equations. An important fact to note is that, while recurrent synaptic connections between neurons in a network give rise to recurrent dynamics, they are not absolutely necessary, as dynamical recurrent can arise without them. This is the case of neurons or synapses which have state which evolve according to internal dynamics: the current state depends on the previous state and the next state depends on the current state, thus state-full units are inherently recurrent. This idea can be very well applied to SNNs, with computations necessary to update a cell state that can be unrolled in time as seen in the figure below.

In the figure: Illustration of the computational graph of a SNN in discrete time. Time steps flow from left to right. Input spikes S(0) are fed into the network from the bottom and propagate upwards to higher layers. The synaptic currents I are decayed by $\alpha$ in each time step and fed into the membrane potentials U. The U are similarly decaying over time as characterized by $\beta$. Spike trains S are generated by applying a threshold non-linearity to the membrane potentials U in each time step. Spikes causally affect the network state (orange connections). First, each spike causes the membrane potential of the neuron that emits the spike to be reset. Second, each spike may be communicated to the same neuronal population via recurrent connections V(1). Finally, it may also be communicated via W(2) to another downstream network layer or, alternatively, a readout layer on which a cost function is defined.

The Backpropagation rule can be applied to RNNs. In this case the recurrence is "unrolled" meaning that an auxiliary network is created by making copies of the network for each time step. The unrolled network is simply a deep network with shared feed-forward weights W(1) and recurrent weights V(1), on which the standard BP applies:

Applying BP to an unrolled network is referred to as Back-Propagation Through Time (BPTT).