Gradient-based optimization powered by the backpropagation algorithm proved to be the pivotal method in the training of non-spiking artificial neural networks. At the same time, spiking neural networks hold the promise for efficient processing of real-world sensory data by communicating using discrete events in continuous time. We derive the backpropagation algorithm for a recurrent network of spiking (leaky integrate-and-fire) neurons with hard thresholds and show that the backward dynamics amount to an event-based backpropagation of errors through time. Our derivation uses the jump conditions for partial derivatives at state discontinuities found by applying the implicit function theorem, allowing us to avoid approximations or substitutions. We find that the gradient exists and is finite almost everywhere in weight space, up to the null set where a membrane potential is precisely tangent to the threshold. Our presented algorithm, EventProp, computes the exact gradient with respect to a general loss function based on spike times and membrane potentials. Crucially, the algorithm allows for an event-based communication scheme in the backward phase, retaining the potential advantages of temporal sparsity afforded by spiking neural networks. We demonstrate the optimization of spiking networks using gradients computed via EventProp and the Yin-Yang and MNIST datasets with either a spike time-based or voltage-based loss function and report competitive performance. Our work supports the rigorous study of gradient-based optimization in spiking neural networks as well as the development of event-based neuromorphic architectures for the efficient training of spiking neural networks. While we consider the leaky integrate-and-fire model in this work, our methodology generalises to any neuron model defined as a hybrid dynamical system.

Neural codes, such as temporal codes (precisely timed spikes) and rate codes (instantaneous spike firing rates), are believed to be used in encoding sensory information into spike trains of cortical neurons. Temporal and rate codes co-exist in the spike train and such multiplexed neural code-carrying spike trains have been shown to be spatially synchronized in multiple neurons across different cortical layers during sensory information processing. Inhibition is suggested to promote such synchronization, but it is unclear whether distinct subtypes of interneurons make different contributions in the synchronization of multiplexed neural codes. To test this, in vivo single-unit recordings from barrel cortex were combined with optogenetic manipulations to determine the contributions of parvalbumin (PV)- and somatostatin (SST)-positive interneurons to synchronization of precisely timed spike sequences. We found that PV interneurons preferentially promote the synchronization of spike times when instantaneous firing rates are low (<12 Hz), whereas SST interneurons preferentially promote the synchronization of spike times when instantaneous firing rates are high (>12 Hz). Furthermore, using a computational model, we demonstrate that these effects can be explained by PV and SST interneurons having preferential contribution to feedforward and feedback inhibition, respectively. Overall, these results show that PV and SST interneurons have distinct frequency (rate code)-selective roles in dynamically gating the synchronization of spike times (temporal code) through preferentially recruiting feedforward and feedback inhibitory circuit motifs. The inhibitory neural circuit mechanisms we uncovered here his may have critical roles in regulating neural code-based somatosensory information processing in the neocortex.

Spike trains recorded from the cortex of behaving animals can be complex, highly variable from trial to trial, and therefore challenging to interpret. A fraction of cells exhibit trial-averaged responses with obvious task-related features such as pure tone frequency tuning in auditory cortex. However, a substantial number of cells (including cells in primary sensory cortex) do not appear to fire in a task-related manner and are often neglected from analysis. We recently used a novel single-trial, spike-timing-based analysis to show that both classically responsive and non-classically responsive cortical neurons contain significant information about sensory stimuli and behavioral decisions suggesting that non-classically responsive cells may play an underappreciated role in perception and behavior. We now expand this investigation to explore the synaptic origins and potential contribution of these cells to network function. To do so, we trained a novel spiking recurrent neural network model that incorporates spike-timing-dependent plasticity (STDP) mechanisms to perform the same task as behaving animals. By leveraging excitatory and inhibitory plasticity rules this model reproduces neurons with response profiles that are consistent with previously published experimental data, including classically responsive and non-classically responsive neurons. We found that both classically responsive and non-classically responsive neurons encode behavioral variables in their spike times as seen in vivo. Interestingly, plasticity in excitatory-to-excitatory synapses increased the proportion of non-classically responsive neurons and may play a significant role in determining response profiles. Finally, our model also makes predictions about the synaptic origins of classically and non-classically responsive neurons which we can compare to in vivo whole-cell recordings taken from the auditory cortex of behaving animals. This approach successfully recapitulates heterogeneous response profiles measured from behaving animals and provides a powerful lens for exploring large-scale neuronal dynamics and the plasticity rules that shape them.

Engineered pattern-recognition systems strive for short time-to-solution and low energy-to-solution characteristics. This represents one of the main driving forces behind the development of neuromorphic devices. For both them and their biological archetypes, this corresponds to using as few spikes as early as possible. The concept of few and early spikes is used as the founding principle in the time-to-first-spike coding scheme. Within this framework, we have developed a spike-timing-based learning algorithm, which we used to train neuronal networks on the mixed-signal neuromorphic platform BrainScaleS-2. We derive, from first principles, error-backpropagation-based learning in networks of leaky integrate-and-fire (LIF) neurons relying only on spike times, for specific configurations of neuronal and synaptic time constants. We explicitly examine applicability to neuromorphic substrates by studying the effects of reduced weight precision and range, as well as of parameter noise. We demonstrate the feasibility of our approach on continuous and discrete data spaces, both in software simulations and on BrainScaleS-2. This narrows the gap between previous models of first-spike-time learning and biological neuronal dynamics and paves the way for fast and energy-efficient neuromorphic applications.

The timing of individual neuronal spikes is essential for biological brains to make fast responses to sensory stimuli. However, conventional artificial neural networks lack the intrinsic temporal coding ability present in biological networks. We propose a spiking neural network model that encodes information in the relative timing of individual neuron spikes. In classification tasks, the output of the network is indicated by the first neuron to spike in the output layer. This temporal coding scheme allows the supervised training of the network with backpropagation, using locally exact derivatives of the postsynaptic spike times with respect to presynaptic spike times. The network operates using a biologically-plausible alpha synaptic transfer function. Additionally, we use trainable synchronisation pulses that provide bias, add flexibility during training and exploit the decay part of the alpha function. We show that such networks can be trained successfully on noisy Boolean logic tasks and on the MNIST dataset encoded in time. The results show that the spiking neural network outperforms comparable spiking models on MNIST and achieves similar quality to fully connected conventional networks with the same architecture. We also find that the spiking network spontaneously discovers two operating regimes, mirroring the accuracy-speed trade-off observed in human decision-making: a slow regime, where a decision is taken after all hidden neurons have spiked and the accuracy is very high, and a fast regime, where a decision is taken very fast but the accuracy is lower. These results demonstrate the computational power of spiking networks with biological characteristics that encode information in the timing of individual neurons. By studying temporal coding in spiking networks, we aim to create building blocks towards energy-efficient and more complex biologically-inspired neural architectures.