In Search of Principles. The quest to understand the neural code presents one of the greatest scientific challenges of our time. Everything we see, hear, taste, and touch is encoded in the chemical and electrical activity of neurons in the brain. These neurons transform sensory stimuli—photons on the retina, vibrations of the cochlea, chemical reactions on the tongue, pressure on the skin—into patterns of neural activity. How are our rich and vibrant sensory experiences encoded in these patterns? Are there general principles underlying this process—laws of neural representation? The goal of theoretical neuroscience, much like theoretical physics, is to uncover these laws. At first glance, the complex and “noisy“ domain of neuroscience may seem patently ill-suited for the rigorous and definitive mathematical analyses of physics. However, an emerging line of empirical results suggest that the mathematically exquisite structure of our physical world may be reflected in its representation in the brain. Indeed, findings from across sensory and motor regions of the brain are beginning to reveal a deep connection between the geometry of the physical world and the structure of the neural code.
Geometry in the Brain. This phenomenon is beautifully illustrated by the head-direction circuit in the fly—the fly's internal compass. As a fly navigates, a circuit of neurons encodes the direction it is heading, keeping track of its orientation like a compass pointer. Remarkably, the neurons in this circuit are connected to each other in a ring-shaped structure that precisely mirrors the geometry of the problem. A bump of neural activity on the ring indicates the fly's heading—the pointer. As the fly changes course, the bump of activity rotates correspondingly. Mathematically, this can be described as equivariance to rotation: A transformation that takes place “out there in the world” is directly mirrored in the neural code. Similar phenomena can be observed throughout brain regions: in the geometrically organized cortical maps of visual features (visual cortex), in the topological structure of the circuit that constructs spatial maps for navigation (entorhinal-hippocampal complex), and in the geometric representation of body movement (motor cortex). This suggests a general computational strategy that is employed throughout the brain to preserve the geometry of data throughout stages of information processing.
Geometry in Deep Neural Networks. Independently but convergently, this very same computational strategy has emerged in the field of deep learning. Equivariant neural networks, a model central to the sub-field of Geometric Deep Learning, preserve the geometry of information as it is processed through each layer. Just as in the head-direction circuit, activity in these networks is equivariant to transformations in the data—”rotating,” for instance, as an input image rotates. This approach yields demonstrable gains in accuracy, efficiency, and robustness, as it permits the network to extrapolate what it has learned under one set of conditions (e.g. an object in a particular pose) to others through a simple transformation of the neural code. Notably, equivariance has also been observed to emerge naturally in generic neural networks that were not explicitly designed for it—an underscore of its universality and importance.
Convergent Evolution in Brains and Machines. The convergence of these strategies in biological and artificial intelligence suggests deep, substrate-agnostic principles for information processing. A fundamental problem that any intelligent system must solve is to build models of the world—simulations of the transformation structure of the environment. Models that capture the environment's dynamics compactly and explicitly—i.e. through equivariance—are powerful tools for robust prediction and generalization. A core hypothesis that I advance in my PhD thesis “A Group Theoretic Framework for Neural Computation” (UC Berkeley, 2021) is that the brain evolved to efficiently encode this transformation structure through the use of group-equivariant representations. Group theory provides the mathematical machinery to precisely describe the geometric transformations that characterize our natural world, such as rotation, scaling, and continuous deformations. Indeed, the mathematics of groups plays a central role in physics and is baked in to the very structure of nature. Group theory was instrumental to the paradigm-shifting unification of models in 20th-century physics. Likewise, I propose, it has the potential to unify our understanding of how neural systems construct world models.