L. F. Abbott

Samuel Lippl, L. F. Abbott, SueYeon Chung

Understanding the asymptotic behavior of gradient-descent training of deep neural networks is essential for revealing inductive biases and improving network performance. We derive the infinite-time training limit of a mathematically tractable class of deep nonlinear neural networks, gated linear networks (GLNs), and generalize these results to gated networks described by general homogeneous polynomials. We study the implications of our results, focusing first on two-layer GLNs. We then apply our theoretical predictions to GLNs trained on MNIST and show how architectural constraints and the implicit bias of gradient descent affect performance. Finally, we show that our theory captures a substantial portion of the inductive bias of ReLU networks. By making the inductive bias explicit, our framework is poised to inform the development of more efficient, biologically plausible, and robust learning algorithms.

Rainer Engelken, Alessandro Ingrosso, Ramin Khajeh et al.

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 unique feature of balanced networks is that, because common external input is dynamically canceled by recurrent feedback, it is far easier to suppress chaos with independent inputs into each neuron than through common input. To study this phenomenon we develop a non-stationary dynamic mean-field theory that determines how the activity statistics and largest Lyapunov exponent depend on frequency and amplitude of the input, recurrent coupling strength, and network size, for both common and independent input. We also show that uncorrelated inputs facilitate learning in balanced networks.

David G. Clark, L. F. Abbott, SueYeon Chung

Backpropagation (BP) uses detailed, unit-specific feedback to train deep neural networks (DNNs) with remarkable success. That biological neural circuits appear to perform credit assignment, but cannot implement BP, implies the existence of other powerful learning algorithms. Here, we explore the extent to which a globally broadcast learning signal, coupled with local weight updates, enables training of DNNs. We present both a learning rule, called global error-vector broadcasting (GEVB), and a class of DNNs, called vectorized nonnegative networks (VNNs), in which this learning rule operates. VNNs have vector-valued units and nonnegative weights past the first layer. The GEVB learning rule generalizes three-factor Hebbian learning, updating each weight by an amount proportional to the inner product of the presynaptic activation and a globally broadcast error vector when the postsynaptic unit is active. We prove that these weight updates are matched in sign to the gradient, enabling accurate credit assignment. Moreover, at initialization, these updates are exactly proportional to the gradient in the limit of infinite network width. GEVB matches the performance of BP in VNNs, and in some cases outperforms direct feedback alignment (DFA) applied in conventional networks. Unlike DFA, GEVB successfully trains convolutional layers. Altogether, our theoretical and empirical results point to a surprisingly powerful role for a global learning signal in training DNNs.

SueYeon Chung, L. F. Abbott

Advances in experimental neuroscience have transformed our ability to explore the structure and function of neural circuits. At the same time, advances in machine learning have unleashed the remarkable computational power of artificial neural networks (ANNs). While these two fields have different tools and applications, they present a similar challenge: namely, understanding how information is embedded and processed through high-dimensional representations to solve complex tasks. One approach to addressing this challenge is to utilize mathematical and computational tools to analyze the geometry of these high-dimensional representations, i.e., neural population geometry. We review examples of geometrical approaches providing insight into the function of biological and artificial neural networks: representation untangling in perception, a geometric theory of classification capacity, disentanglement and abstraction in cognitive systems, topological representations underlying cognitive maps, dynamic untangling in motor systems, and a dynamical approach to cognition. Together, these findings illustrate an exciting trend at the intersection of machine learning, neuroscience, and geometry, in which neural population geometry provides a useful population-level mechanistic descriptor underlying task implementation. Importantly, geometric descriptions are applicable across sensory modalities, brain regions, network architectures and timescales. Thus, neural population geometry has the potential to unify our understanding of structure and function in biological and artificial neural networks, bridging the gap between single neurons, populations and behavior.

Alessandro Ingrosso, L. F. Abbott

The construction of biologically plausible models of neural circuits is crucial for understanding the computational properties of the nervous system. Constructing functional networks composed of separate excitatory and inhibitory neurons obeying Dale's law presents a number of challenges. We show how a target-based approach, when combined with a fast online constrained optimization technique, is capable of building functional models of rate and spiking recurrent neural networks in which excitation and inhibition are balanced. Balanced networks can be trained to produce complicated temporal patterns and to solve input-output tasks while retaining biologically desirable features such as Dale's law and response variability.

Theodore H. Moskovitz, Ashok Litwin-Kumar, L. F. Abbott

Ongoing studies have identified similarities between neural representations in biological networks and in deep artificial neural networks. This has led to renewed interest in developing analogies between the backpropagation learning algorithm used to train artificial networks and the synaptic plasticity rules operative in the brain. These efforts are challenged by biologically implausible features of backpropagation, one of which is a reliance on symmetric forward and backward synaptic weights. A number of methods have been proposed that do not rely on weight symmetry but, thus far, these have failed to scale to deep convolutional networks and complex data. We identify principal obstacles to the scalability of such algorithms and introduce several techniques to mitigate them. We demonstrate that a modification of the feedback alignment method that enforces a weaker form of weight symmetry, one that requires agreement of weight sign but not magnitude, can achieve performance competitive with backpropagation. Our results complement those of Bartunov et al. (2018) and Xiao et al. (2018b) and suggest that mechanisms that promote alignment of feedforward and feedback weights are critical for learning in deep networks.

Brian DePasquale, Christopher J. Cueva, Kanaka Rajan et al.

Trained recurrent networks are powerful tools for modeling dynamic neural computations. We present a target-based method for modifying the full connectivity matrix of a recurrent network to train it to perform tasks involving temporally complex input/output transformations. The method introduces a second network during training to provide suitable "target" dynamics useful for performing the task. Because it exploits the full recurrent connectivity, the method produces networks that perform tasks with fewer neurons and greater noise robustness than traditional least-squares (FORCE) approaches. In addition, we show how introducing additional input signals into the target-generating network, which act as task hints, greatly extends the range of tasks that can be learned and provides control over the complexity and nature of the dynamics of the trained, task-performing network.

Ran Rubin, L. F. Abbott, Haim Sompolinsky

Neurons and networks in the cerebral cortex must operate reliably despite multiple sources of noise. To evaluate the impact of both input and output noise, we determine the robustness of single-neuron stimulus selective responses, as well as the robustness of attractor states of networks of neurons performing memory tasks. We find that robustness to output noise requires synaptic connections to be in a balanced regime in which excitation and inhibition are strong and largely cancel each other. We evaluate the conditions required for this regime to exist and determine the properties of networks operating within it. A plausible synaptic plasticity rule for learning that balances weight configurations is presented. Our theory predicts an optimal ratio of the number of excitatory and inhibitory synapses for maximizing the encoding capacity of balanced networks for a given statistics of afferent activations. Previous work has shown that balanced networks amplify spatio-temporal variability and account for observed asynchronous irregular states. Here we present a novel type of balanced network that amplifies small changes in the impinging signals, and emerges automatically from learning to perform neuronal and network functions robustly.

David Sussillo, Rafal Jozefowicz, L. F. Abbott et al.

Neuroscience is experiencing a data revolution in which many hundreds or thousands of neurons are recorded simultaneously. Currently, there is little consensus on how such data should be analyzed. Here we introduce LFADS (Latent Factor Analysis via Dynamical Systems), a method to infer latent dynamics from simultaneously recorded, single-trial, high-dimensional neural spiking data. LFADS is a sequential model based on a variational auto-encoder. By making a dynamical systems hypothesis regarding the generation of the observed data, LFADS reduces observed spiking to a set of low-dimensional temporal factors, per-trial initial conditions, and inferred inputs. We compare LFADS to existing methods on synthetic data and show that it significantly out-performs them in inferring neural firing rates and latent dynamics.

David Sussillo, L. F. Abbott

Training very deep networks is an important open problem in machine learning. One of many difficulties is that the norm of the back-propagated error gradient can grow or decay exponentially. Here we show that training very deep feed-forward networks (FFNs) is not as difficult as previously thought. Unlike when back-propagation is applied to a recurrent network, application to an FFN amounts to multiplying the error gradient by a different random matrix at each layer. We show that the successive application of correctly scaled random matrices to an initial vector results in a random walk of the log of the norm of the resulting vectors, and we compute the scaling that makes this walk unbiased. The variance of the random walk grows only linearly with network depth and is inversely proportional to the size of each layer. Practically, this implies a gradient whose log-norm scales with the square root of the network depth and shows that the vanishing gradient problem can be mitigated by increasing the width of the layers. Mathematical analyses and experimental results using stochastic gradient descent to optimize tasks related to the MNIST and TIMIT datasets are provided to support these claims. Equations for the optimal matrix scaling are provided for the linear and ReLU cases.