Spiking Neural Networks (SNNs) are biologically-inspired models that are capable of processing information in streams of action potentials. However, simulating and training SNNs is computationally expensive due to the need to solve large systems of coupled differential equations. In this paper, we propose a novel event-based algorithm called SparseProp for simulating and training sparse SNNs. Our algorithm reduces the computational cost of both forward pass and backward pass operations from O(N) to O(log(N)) per network spike, enabling numerically exact simulations of large spiking networks and their efficient training using backpropagation through time. By exploiting the sparsity of the network, SparseProp avoids iterating through all neurons at every spike and uses efficient state updates. We demonstrate the effectiveness of SparseProp for several classical integrate-and-fire neuron models, including simulating a sparse SNN with one million LIF neurons, which is sped up by more than four orders of magnitude compared to previous implementations. Our work provides an efficient and exact solution for training large-scale spiking neural networks and opens up new possibilities for building more sophisticated brain-inspired models.

Training recurrent neural networks (RNNs) remains a challenge due to the instability of gradients across long time horizons, which can lead to exploding and vanishing gradients. Recent research has linked these problems to the values of Lyapunov exponents for the forward-dynamics, which describe the growth or shrinkage of infinitesimal perturbations. Here, we propose gradient flossing, a novel approach to tackling gradient instability by pushing Lyapunov exponents of the forward dynamics toward zero during learning. We achieve this by regularizing Lyapunov exponents through backpropagation using differentiable linear algebra. This enables us to "floss" the gradients, stabilizing them and thus improving network training. We show that gradient flossing controls not only the gradient norm but also the condition number of the long-term Jacobian, facilitating multidimensional error feedback propagation. We find that applying gradient flossing before training enhances both the success rate and convergence speed for tasks involving long time horizons. For challenging tasks, we show that gradient flossing during training can further increase the time horizon that can be bridged by backpropagation through time. Moreover, we demonstrate the effectiveness of our approach on various RNN architectures and tasks of variable temporal complexity. Additionally, we provide a simple implementation of our gradient flossing algorithm that can be used in practice. Our results indicate that gradient flossing via regularizing Lyapunov exponents can significantly enhance the effectiveness of RNN training and mitigate the exploding and vanishing gradients problem.

Information encoding in neural circuits depends on how well time-varying stimuli are encoded by neural populations.Slow neuronal timescales, noise and network chaos can compromise reliable and rapid population response to external stimuli.A dynamic balance of externally incoming currents by strong recurrent inhibition was previously proposed as a mechanism to accurately and robustly encode a time-varying stimulus in balanced networks of binary neurons, but a theory for recurrent rate networks was missing. Here, we develop a non-stationary dynamic mean-field theory that transparently explains how a tight balance of excitatory currents by recurrent inhibition improves information encoding. We demonstrate that the mutual information rate of a time-varying input increases linearly with the tightness of balance, both in the presence of additive noise and with recurrently generated chaotic network fluctuations. We corroborated our findings in numerical experiments and demonstrated that recurrent networks with positive firing rates trained to transmit a time-varying stimulus generically use recurrent inhibition to increase the information rate. We also found that networks trained to transmit multiple independent time-varying signals spontaneously form multiple local inhibitory clusters, one for each input channel.Our findings suggest that feedforward excitatory input and local recurrent inhibition - as observed in many biological circuits - is a generic circuit motif for encoding and transmitting time-varying information in recurrent neural circuits.

Neural circuits exhibit complex activity patterns, both spontaneously and evoked by external stimuli. Information encoding and learning in neural circuits depend on how well time-varying stimuli can control spontaneous network activity. We show that in firing-rate networks in the balanced state, external control of recurrent dynamics, i.e., the suppression of internally-generated chaotic variability, strongly depends on correlations in the input. A distinctive feature of balanced networks is that, because common external input is dynamically canceled by recurrent feedback, it is far more difficult to suppress chaos with common input into each neuron than through independent input. To study this phenomenon, we develop a non-stationary dynamic mean-field theory for driven networks. The theory explains how the activity statistics and the largest Lyapunov exponent depend on the frequency and amplitude of the input, recurrent coupling strength, and network size, for both common and independent input. We further show that uncorrelated inputs facilitate learning in balanced networks.