Advancing mathematics by guiding human intuition with AI.

The practice of mathematics involves discovering patterns and using these to formulate and prove conjectures, resulting in theorems. Since the 1960s, mathematicians have used computers to assist in the discovery of patterns and formulation of conjectures1, most famously in the Birch and Swinnerton-Dyer conjecture2, a Millennium Prize Problem3. Here we provide examples of new fundamental results in pure mathematics that have been discovered with the assistance of machine learning-demonstrating a method by which machine learning can aid mathematicians in discovering new conjectures and theorems. We propose a process of using machine learning to discover potential patterns and relations between mathematical objects, understanding them with attribution techniques and using these observations to guide intuition and propose conjectures. We outline this machine-learning-guided framework and demonstrate its successful application to current research questions in distinct areas of pure mathematics, in each case showing how it led to meaningful mathematical contributions on important open problems: a new connection between the algebraic and geometric structure of knots, and a candidate algorithm predicted by the combinatorial invariance conjecture for symmetric groups4. Our work may serve as a model for collaboration between the fields of mathematics and artificial intelligence (AI) that can achieve surprising results by leveraging the respective strengths of mathematicians and machine learning.

Bayesian computation emerges in generic cortical microcircuits through spike-timing-dependent plasticity.

The principles by which networks of neurons compute, and how spike-timing dependent plasticity (STDP) of synaptic weights generates and maintains their computational function, are unknown. Preceding work has shown that soft winner-take-all (WTA) circuits, where pyramidal neurons inhibit each other via interneurons, are a common motif of cortical microcircuits. We show through theoretical analysis and computer simulations that Bayesian computation is induced in these network motifs through STDP in combination with activity-dependent changes in the excitability of neurons. The fundamental components of this emergent Bayesian computation are priors that result from adaptation of neuronal excitability and implicit generative models for hidden causes that are created in the synaptic weights through STDP. In fact, a surprising result is that STDP is able to approximate a powerful principle for fitting such implicit generative models to high-dimensional spike inputs: Expectation Maximization. Our results suggest that the experimentally observed spontaneous activity and trial-to-trial variability of cortical neurons are essential features of their information processing capability, since their functional role is to represent probability distributions rather than static neural codes. Furthermore it suggests networks of Bayesian computation modules as a new model for distributed information processing in the cortex.

Neural dynamics as sampling: a model for stochastic computation in recurrent networks of spiking neurons.

The organization of computations in networks of spiking neurons in the brain is still largely unknown, in particular in view of the inherently stochastic features of their firing activity and the experimentally observed trial-to-trial variability of neural systems in the brain. In principle there exists a powerful computational framework for stochastic computations, probabilistic inference by sampling, which can explain a large number of macroscopic experimental data in neuroscience and cognitive science. But it has turned out to be surprisingly difficult to create a link between these abstract models for stochastic computations and more detailed models of the dynamics of networks of spiking neurons. Here we create such a link and show that under some conditions the stochastic firing activity of networks of spiking neurons can be interpreted as probabilistic inference via Markov chain Monte Carlo (MCMC) sampling. Since common methods for MCMC sampling in distributed systems, such as Gibbs sampling, are inconsistent with the dynamics of spiking neurons, we introduce a different approach based on non-reversible Markov chains that is able to reflect inherent temporal processes of spiking neuronal activity through a suitable choice of random variables. We propose a neural network model and show by a rigorous theoretical analysis that its neural activity implements MCMC sampling of a given distribution, both for the case of discrete and continuous time. This provides a step towards closing the gap between abstract functional models of cortical computation and more detailed models of networks of spiking neurons.

Probabilistic inference in general graphical models through sampling in stochastic networks of spiking neurons.

An important open problem of computational neuroscience is the generic organization of computations in networks of neurons in the brain. We show here through rigorous theoretical analysis that inherent stochastic features of spiking neurons, in combination with simple nonlinear computational operations in specific network motifs and dendritic arbors, enable networks of spiking neurons to carry out probabilistic inference through sampling in general graphical models. In particular, it enables them to carry out probabilistic inference in Bayesian networks with converging arrows ("explaining away") and with undirected loops, that occur in many real-world tasks. Ubiquitous stochastic features of networks of spiking neurons, such as trial-to-trial variability and spontaneous activity, are necessary ingredients of the underlying computational organization. We demonstrate through computer simulations that this approach can be scaled up to neural emulations of probabilistic inference in fairly large graphical models, yielding some of the most complex computations that have been carried out so far in networks of spiking neurons.

Learning stable, regularised latent models of neural population dynamics.

Ongoing advances in experimental technique are making commonplace simultaneous recordings of the activity of tens to hundreds of cortical neurons at high temporal resolution. Latent population models, including Gaussian-process factor analysis and hidden linear dynamical system (LDS) models, have proven effective at capturing the statistical structure of such data sets. They can be estimated efficiently, yield useful visualisations of population activity, and are also integral building-blocks of decoding algorithms for brain-machine interfaces (BMI). One practical challenge, particularly to LDS models, is that when parameters are learned using realistic volumes of data the resulting models often fail to reflect the true temporal continuity of the dynamics; and indeed may describe a biologically-implausible unstable population dynamic that is, it may predict neural activity that grows without bound. We propose a method for learning LDS models based on expectation maximisation that constrains parameters to yield stable systems and at the same time promotes capture of temporal structure by appropriate regularisation. We show that when only little training data is available our method yields LDS parameter estimates which provide a substantially better statistical description of the data than alternatives, whilst guaranteeing stable dynamics. We demonstrate our methods using both synthetic data and extracellular multi-electrode recordings from motor cortex.

A spiking neuron as information bottleneck.

Neurons receive thousands of presynaptic input spike trains while emitting a single output spike train. This drastic dimensionality reduction suggests considering a neuron as a bottleneck for information transmission. Extending recent results, we propose a simple learning rule for the weights of spiking neurons derived from the information bottleneck (IB) framework that minimizes the loss of relevant information transmitted in the output spike train. In the IB framework, relevance of information is defined with respect to contextual information, the latter entering the proposed learning rule as a "third" factor besides pre- and postsynaptic activities. This renders the theoretically motivated learning rule a plausible model for experimentally observed synaptic plasticity phenomena involving three factors. Furthermore, we show that the proposed IB learning rule allows spiking neurons to learn a predictive code, that is, to extract those parts of their input that are predictive for future input.

Neural scene representation and rendering.

Scene representation-the process of converting visual sensory data into concise descriptions-is a requirement for intelligent behavior. Recent work has shown that neural networks excel at this task when provided with large, labeled datasets. However, removing the reliance on human labeling remains an important open problem. To this end, we introduce the Generative Query Network (GQN), a framework within which machines learn to represent scenes using only their own sensors. The GQN takes as input images of a scene taken from different viewpoints, constructs an internal representation, and uses this representation to predict the appearance of that scene from previously unobserved viewpoints. The GQN demonstrates representation learning without human labels or domain knowledge, paving the way toward machines that autonomously learn to understand the world around them.

Distributed Bayesian Computation and Self-Organized Learning in Sheets of Spiking Neurons with Local Lateral Inhibition.

During the last decade, Bayesian probability theory has emerged as a framework in cognitive science and neuroscience for describing perception, reasoning and learning of mammals. However, our understanding of how probabilistic computations could be organized in the brain, and how the observed connectivity structure of cortical microcircuits supports these calculations, is rudimentary at best. In this study, we investigate statistical inference and self-organized learning in a spatially extended spiking network model, that accommodates both local competitive and large-scale associative aspects of neural information processing, under a unified Bayesian account. Specifically, we show how the spiking dynamics of a recurrent network with lateral excitation and local inhibition in response to distributed spiking input, can be understood as sampling from a variational posterior distribution of a well-defined implicit probabilistic model. This interpretation further permits a rigorous analytical treatment of experience-dependent plasticity on the network level. Using machine learning theory, we derive update rules for neuron and synapse parameters which equate with Hebbian synaptic and homeostatic intrinsic plasticity rules in a neural implementation. In computer simulations, we demonstrate that the interplay of these plasticity rules leads to the emergence of probabilistic local experts that form distributed assemblies of similarly tuned cells communicating through lateral excitatory connections. The resulting sparse distributed spike code of a well-adapted network carries compressed information on salient input features combined with prior experience on correlations among them. Our theory predicts that the emergence of such efficient representations benefits from network architectures in which the range of local inhibition matches the spatial extent of pyramidal cells that share common afferent input.

Democratic population decisions result in robust policy-gradient learning: a parametric study with GPU simulations.

High performance computing on the Graphics Processing Unit (GPU) is an emerging field driven by the promise of high computational power at a low cost. However, GPU programming is a non-trivial task and moreover architectural limitations raise the question of whether investing effort in this direction may be worthwhile. In this work, we use GPU programming to simulate a two-layer network of Integrate-and-Fire neurons with varying degrees of recurrent connectivity and investigate its ability to learn a simplified navigation task using a policy-gradient learning rule stemming from Reinforcement Learning. The purpose of this paper is twofold. First, we want to support the use of GPUs in the field of Computational Neuroscience. Second, using GPU computing power, we investigate the conditions under which the said architecture and learning rule demonstrate best performance. Our work indicates that networks featuring strong Mexican-Hat-shaped recurrent connections in the top layer, where decision making is governed by the formation of a stable activity bump in the neural population (a "non-democratic" mechanism), achieve mediocre learning results at best. In absence of recurrent connections, where all neurons "vote" independently ("democratic") for a decision via population vector readout, the task is generally learned better and more robustly. Our study would have been extremely difficult on a desktop computer without the use of GPU programming. We present the routines developed for this purpose and show that a speed improvement of 5x up to 42x is provided versus optimised Python code. The higher speed is achieved when we exploit the parallelism of the GPU in the search of learning parameters. This suggests that efficient GPU programming can significantly reduce the time needed for simulating networks of spiking neurons, particularly when multiple parameter configurations are investigated.

Spike-frequency adapting neural ensembles: beyond mean adaptation and renewal theories.

We propose a Markov process model for spike-frequency adapting neural ensembles that synthesizes existing mean-adaptation approaches, population density methods, and inhomogeneous renewal theory, resulting in a unified and tractable framework that goes beyond renewal and mean-adaptation theories by accounting for correlations between subsequent interspike intervals. A method for efficiently generating inhomogeneous realizations of the proposed Markov process is given, numerical methods for solving the population equation are presented, and an expression for the first-order interspike interval correlation is derived. Further, we show that the full five-dimensional master equation for a conductance-based integrate-and-fire neuron with spike-frequency adaptation and a relative refractory mechanism driven by Poisson spike trains can be reduced to a two-dimensional generalization of the proposed Markov process by an adiabatic elimination of fast variables. For static and dynamic stimulation, negative serial interspike interval correlations and transient population responses, respectively, of Monte Carlo simulations of the full five-dimensional system can be accurately described by the proposed two-dimensional Markov process.