Networks based on coordinated spike coding can encode information with high efficiency in the spike trains of individual neurons. These networks exhibit single-neuron variability and tuning curves as typically observed in cortex, but paradoxically coincide with a precise, non-redundant spike-based population code. However, it has remained unclear whether the specific synaptic connectivities required in these networks can be learnt with local learning rules. Here, we show how to learn the required architecture. Using coding efficiency as an objective, we derive spike-timing-dependent learning rules for a recurrent neural network, and we provide exact solutions for the networks' convergence to an optimal state. As a result, we deduce an entire network from its input distribution and a firing cost. After learning, basic biophysical quantities such as voltages, firing thresholds, excitation, inhibition, or spikes acquire precise functional interpretations.

To predict sensory inputs or control motor trajectories, the brain must constantly learn temporal dynamics based on error feedback. However, it remains unclear how such supervised learning is implemented in biological neural networks. Learning in recurrent spiking networks is notoriously difficult because local changes in connectivity may have an unpredictable effect on the global dynamics. The most commonly used learning rules, such as temporal back-propagation, are not local and thus not biologically plausible. Furthermore, reproducing the Poisson-like statistics of neural responses requires the use of networks with balanced excitation and inhibition. Such balance is easily destroyed during learning. Using a top-down approach, we show how networks of integrate-and-fire neurons can learn arbitrary linear dynamical systems by feeding back their error as a feed-forward input. The network uses two types of recurrent connections: fast and slow. The fast connections learn to balance excitation and inhibition using a voltage-based plasticity rule. The slow connections are trained to minimize the error feedback using a current-based Hebbian learning rule. Importantly, the balance maintained by fast connections is crucial to ensure that global error signals are available locally in each neuron, in turn resulting in a local learning rule for the slow connections. This demonstrates that spiking networks can learn complex dynamics using purely local learning rules, using E/I balance as the key rather than an additional constraint. The resulting network implements a given function within the predictive coding scheme, with minimal dimensions and activity.

How do neural networks learn to represent information? Here, we address this question by assuming that neural networks seek to generate an optimal population representation for a fixed linear decoder. We define a loss function for the quality of the population read-out and derive the dynamical equations for both neurons and synapses from the requirement to minimize this loss. The dynamical equations yield a network of integrate-and-fire neurons undergoing Hebbian plasticity. We show that, through learning, initially regular and highly correlated spike trains evolve towards Poisson-distributed and independent spike trains with much lower firing rates. The learning rule drives the network into an asynchronous, balanced regime where all inputs to the network are represented optimally for the given decoder. We show that the network dynamics and synaptic plasticity jointly balance the excitation and inhibition received by each unit as tightly as possible and, in doing so, minimize the prediction error between the inputs and the decoded outputs. In turn, spikes are only signalled whenever this prediction error exceeds a certain value, thereby implementing a predictive coding scheme. Our work suggests that several of the features reported in cortical networks, such as the high trial-to-trial variability, the balance between excitation and inhibition, and spike-timing dependent plasticity, are simply signatures of an efficient, spike-based code.