Recurrent spiking neural networks (RSNN) in the  brain learn to perform a wide range of perceptual, cognitive and motor tasks very efficiently in terms of time and energy consumption . Moreover, learning can happen rapidly after very few examples. This is due to the optimality of coding and learning schemes, which have yet to be clearly understood. This naturally challenges the formulation of biologically inspired RSNNs in order to improve our understanding of biological intelligence and the efficiency of artificial ones . Several spiking network models have been proposed  but it remains a challenge to design RSNNs that use biologically plausible mechanisms  capable of solving complex temporal tasks. We use a general probabilistic framework that relies on the principle of maximizing the likelihood for the network to reproduce some temporal dynamics. This principle permits to analytically work out an explicit and biologically plausible plasticity rule. Here we propose a novel target-based learning scheme in which such a rule can be used to efficiently train a RSNN to solve several temporal taskssuch as learning multidimensional trajectory and an implementation of the temporal XOR. We finally 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 is general and it can be applied to a wide variety of network architectures and types of biological neurons. The  derived plasticity learning rule is specific to each neuron model and can produce a theoretical prediction to be experimentally verified .
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 .
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reproduce some temporal dynamics. This principle permits to analytically work out an explicit and biologically plausible plasticity rule. Here we
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such a rule can be used to efficiently train a RSNN to solve several temporal taskssuch as learning multidimensional trajectory and an implementation of the temporal XOR. We finally
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
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