Seminars

What is the relationship between task, network architecture, and population activity in nonlinear deep networks? I will describe the Gated Deep Linear Network framework, which schematizes how pathways of information flow impact learning dynamics within an architecture. Because of the gating, these networks can compute nonlinear functions of their input. We derive an exact reduction and, for certain cases, exact solutions to the dynamics of learning. The reduction takes the form of a neural race with an implicit bias towards shared representations, which then govern the model’s ability to systematically generalize, multi-task, and transfer. We show how appropriate network architectures can help factorize and abstract knowledge. Together, these results begin to shed light on the links between architecture, learning dynamics and network performance.

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.

The success of deep learning sparked interest in whether the brain learns by using similar techniques for assigning credit to each synaptic weight for its contribution to the network output. However, the majority of current attempts at biologically-plausible learning methods are either non-local in time, require highly specific connectivity motives, or have no clear link to any known mathematical optimization method. Here, we introduce Deep Feedback Control (DFC), a new learning method that uses a feedback controller to drive a deep neural network to match a desired output target and whose control signal can be used for credit assignment. The resulting learning rule is fully local in space and time and approximates Gauss-Newton optimization for a wide range of feedback connectivity patterns. To further underline its biological plausibility, we relate DFC to a multi-compartment model of cortical pyramidal neurons with a local voltage-dependent synaptic plasticity rule, consistent with recent theories of dendritic processing. By combining dynamical system theory with mathematical optimization theory, we provide a strong theoretical foundation for DFC that we corroborate with detailed results on toy experiments and standard computer-vision benchmarks.

Rajanlab designs neural network models constrained by experimental data, and reverse engineers them to figure out how brain circuits function in health and disease. Recently, we have been developing a powerful new theory-based framework for “in-vivo tract tracing” from multi-regional neural activity collected experimentally. We call this framework CURrent-Based Decomposition (CURBD). CURBD employs recurrent neural networks (RNNs) directly constrained, from the outset, by time series measurements acquired experimentally, such as Ca2+ imaging or electrophysiological data. Once trained, these data-constrained RNNs let us infer matrices quantifying the interactions between all pairs of modeled units. Such model-derived “directed interaction matrices” can then be used to separately compute excitatory and inhibitory input currents that drive a given neuron from all other neurons. Therefore different current sources can be de-mixed – either within the same region or from other regions, potentially brain-wide – which collectively give rise to the population dynamics observed experimentally. Source de-mixed currents obtained through CURBD allow an unprecedented view into multi-region mechanisms inaccessible from measurements alone. We have applied this method successfully to several types of neural data from our experimental collaborators, e.g., zebrafish (Deisseroth lab, Stanford), mice (Harvey lab, Harvard), monkeys (Rudebeck lab, Sinai), and humans (Rutishauser lab, Cedars Sinai), where we have discovered both directed interactions brain wide and inter-area currents during different types of behaviors. With this powerful framework based on data-constrained multi-region RNNs and CURrent Based Decomposition (CURBD), we ask if there are conserved multi-region mechanisms across different species, as well as identify key divergences.