  Recurrent spiking neural networks (RSNN) in the human brain learn to perform
a wide range of perceptual, cognitive and motor tasks very efficiently in terms
of energy consumption and requires very few examples. This motivates the search
for biologically inspired learning rules for RSNNs to improve our understanding
of brain computation and the efficiency of artificial intelligence. Several
spiking models and learning rules have been proposed, but it remains a
challenge to design RSNNs whose learning relies on biologically plausible
mechanisms and are capable of solving complex temporal tasks. In this paper, we
derive a learning rule, local to the synapse, from a simple mathematical
principle, the maximization of the likelihood for the network to solve a
specific task. We propose a novel target-based learning scheme in which the
learning rule derived from likelihood maximization is used to mimic a specific
spiking pattern that encodes the solution to complex temporal tasks. This
method makes the learning extremely rapid and precise, outperforming state of
the art algorithms for RSNNs. We demonstrate the capacity of our model to
tackle several problems like learning multidimensional trajectories and solving
the classical temporal XOR benchmark. Finally, we show that an online
approximation of the gradient ascent, in addition to guaranteeing complete
locality in time and space, allows learning after very few presentations of the
target output. Our model can be applied to different types of biological
neurons. The analytically derived plasticity learning rule is specific to each
neuron model and can produce a theoretical prediction for experimental
validation.
