To accurately track self-location, animals need to integrate their movements through space. In amniotes, representations of self-location have been found in regions such as the hippocampus. It is unknown whether more ancient brain regions contain such representations and by which pathways they may drive locomotion. Fish displaced by water currents must prevent uncontrolled drift to potentially dangerous areas. We found that larval zebrafish track such movements and can later swim back to their earlier location. Whole-brain functional imaging revealed the circuit enabling this process of positional homeostasis. Position-encoding brainstem neurons integrate optic flow, then bias future swimming to correct for past displacements by modulating inferior olive and cerebellar activity. Manipulation of position-encoding or olivary neurons abolished positional homeostasis or evoked behavior as if animals had experienced positional shifts. These results reveal a multiregional hindbrain circuit in vertebrates for optic flow integration, memory of self-location, and its neural pathway to behavior.Competing Interest StatementThe authors have declared no competing interest.

Neural computation in biological and artificial networks relies on nonlinear synaptic integration. The structural connectivity matrix of synaptic weights between neurons is a critical determinant of overall network function. However, quantitative links between neural network structure and function are complex and subtle. For example, many networks can give rise to similar functional responses, and the same network can function differently depending on context. Whether certain patterns of synaptic connectivity are required to generate specific network-level computations is largely unknown. Here we introduce a geometric framework for identifying synaptic connections required by steady-state responses in recurrent networks of rectified-linear neurons. Assuming that the number of specified response patterns does not exceed the number of input synapses, we analytically calculate all feedforward and recurrent connectivity matrices that can generate the specified responses from the network inputs. We then use this analytical characterization to rigorously analyze the solution space geometry and derive certainty conditions guaranteeing a non-zero synapse between neurons. Numerical simulations of feedforward and recurrent networks verify our analytical results. Our theoretical framework could be applied to neural activity data to make anatomical predictions that follow generally from the model architecture. It thus provides novel opportunities for discerning what model features are required to accurately relate neural network structure and function.

All animals must transform ambiguous sensory data into successful behavior. This requires sensory representations that accurately reflect the statistics of natural stimuli and behavior. Multiple studies show that visual motion processing is tuned for accuracy under naturalistic conditions, but the sensorimotor circuits extracting these cues and implementing motion-guided behavior remain unclear. Here we show that the larval zebrafish retina extracts a diversity of naturalistic motion cues, and the retinorecipient pretectum organizes these cues around the elements of behavior. We find that higher-order motion stimuli, gliders, induce optomotor behavior matching expectations from natural scene analyses. We then image activity of retinal ganglion cell terminals and pretectal neurons. The retina exhibits direction-selective responses across glider stimuli, and anatomically clustered pretectal neurons respond with magnitudes matching behavior. Peripheral computations thus reflect natural input statistics, whereas central brain activity precisely codes information needed for behavior. This general principle could organize sensorimotor transformations across animal species.

Animals detect motion using a variety of visual cues that reflect regularities in the natural world. Experiments in animals across phyla have shown that motion percepts incorporate both pairwise and triplet spatiotemporal correlations that could theoretically benefit motion computation. However, it remains unclear how visual systems assemble these cues to build accurate motion estimates. Here we used systematic behavioral measurements of fruit fly motion perception to show how flies combine local pairwise and triplet correlations to reduce variability in motion estimates across natural scenes. By generating synthetic images with statistics controlled by maximum entropy distributions, we show that the triplet correlations are useful only when images have light-dark asymmetries that mimic natural ones. This suggests that asymmetric ON-OFF processing is tuned to the particular statistics of natural scenes. Since all animals encounter the world's light-dark asymmetries, many visual systems are likely to use asymmetric ON-OFF processing to improve motion estimation.

Breakthrough technologies for monitoring and manipulating single-neuron activity provide unprecedented opportunities for whole-brain neuroscience in larval zebrafish1–9. Understanding the neural mechanisms of visually guided behavior also requires precise stimulus control, but little prior research has accounted for physical distortions that result from refraction and reflection at an air-water interface that usually separates the projected stimulus from the fish10–12. Here we provide a computational tool that transforms between projected and received stimuli in order to detect and control these distortions. The tool considers the most commonly encountered interface geometry, and we show that this and other common configurations produce stereotyped distortions. By correcting these distortions, we reduced discrepancies in the literature concerning stimuli that evoke escape behavior13,14, and we expect this tool will help reconcile other confusing aspects of the literature. This tool also aids experimental design, and we illustrate the dangers that uncorrected stimuli pose to receptive field mapping experiments.

Learning in deep neural networks is known to depend critically on the knowledge embedded in the initial network weights. However, few theoretical results have precisely linked prior knowledge to learning dynamics. Here we derive exact solutions to the dynamics of learning with rich prior knowledge in deep linear networks by generalising Fukumizu's matrix Riccati solution \citep{fukumizu1998effect}. We obtain explicit expressions for the evolving network function, hidden representational similarity, and neural tangent kernel over training for a broad class of initialisations and tasks. The expressions reveal a class of task-independent initialisations that radically alter learning dynamics from slow non-linear dynamics to fast exponential trajectories while converging to a global optimum with identical representational similarity, dissociating learning trajectories from the structure of initial internal representations. We characterise how network weights dynamically align with task structure, rigorously justifying why previous solutions successfully described learning from small initial weights without incorporating their fine-scale structure. Finally, we discuss the implications of these findings for continual learning, reversal learning and learning of structured knowledge. Taken together, our results provide a mathematical toolkit for understanding the impact of prior knowledge on deep learning.

A cognitive compass enabling spatial navigation requires neural representation of heading direction (HD), yet the neural circuit architecture enabling this representation remains unclear. While various network models have been proposed to explain HD systems, these models rely on simplified circuit architectures that are incompatible with empirical observations from connectomes. Here we construct a novel network model for the fruit fly HD system that satisfies both connectome-derived architectural constraints and the functional requirement of continuous heading representation. We characterize an ensemble of continuous attractor networks where compass neurons providing local mutual excitation are coupled to inhibitory neurons. We discover a new mechanism where continuous heading representation emerges from combining symmetric and anti-symmetric activity patterns. Our analysis reveals three distinct realizations of these networks that all match observed compass neuron activity but differ in their predictions for inhibitory neuron activation patterns. Further, we found that deviations from these realizations can be compensated by cell-type-specific rescaling of synaptic weights, which could be potentially achieved through neuromodulation. This framework can be extended to incorporate the complete fly central complex connectome and could reveal principles of neural circuits representing other continuous quantities, such as spatial location, across insects and vertebrates.

Cognitive maps confer animals with flexible intelligence by representing spatial, temporal and abstract relationships that can be used to shape thought, planning and behaviour. Cognitive maps have been observed in the hippocampus1, but their algorithmic form and learning mechanisms remain obscure. Here we used large-scale, longitudinal two-photon calcium imaging to record activity from thousands of neurons in the CA1 region of the hippocampus while mice learned to efficiently collect rewards from two subtly different linear tracks in virtual reality. Throughout learning, both animal behaviour and hippocampal neural activity progressed through multiple stages, gradually revealing improved task representation that mirrored improved behavioural efficiency. The learning process involved progressive decorrelations in initially similar hippocampal neural activity within and across tracks, ultimately resulting in orthogonalized representations resembling a state machine capturing the inherent structure of the task. This decorrelation process was driven by individual neurons acquiring task-state-specific responses (that is, 'state cells'). Although various standard artificial neural networks did not naturally capture these dynamics, the clone-structured causal graph, a hidden Markov model variant, uniquely reproduced both the final orthogonalized states and the learning trajectory seen in animals. The observed cellular and population dynamics constrain the mechanisms underlying cognitive map formation in the hippocampus, pointing to hidden state inference as a fundamental computational principle, with implications for both biological and artificial intelligence.

Understanding learning through synaptic plasticity rules in the brain is a grand challenge for neuroscience. Here we introduce a novel computational framework for inferring plasticity rules from experimental data on neural activity trajectories and behavioral learning dynamics. Our methodology parameterizes the plasticity function to provide theoretical interpretability and facilitate gradient-based optimization. For instance, we use Taylor series expansions or multilayer perceptrons to approximate plasticity rules, and we adjust their parameters via gradient descent over entire trajectories to closely match observed neural activity and behavioral data. Notably, our approach can learn intricate rules that induce long nonlinear time-dependencies, such as those incorporating postsynaptic activity and current synaptic weights. We validate our method through simulations, accurately recovering established rules, like Oja’s, as well as more complex hypothetical rules incorporating reward-modulated terms. We assess the resilience of our technique to noise and, as a tangible application, apply it to behavioral data from Drosophila during a probabilistic reward-learning experiment. Remarkably, we identify an active forgetting component of reward learning in flies that enhances the predictive accuracy of previous models. Overall, our modeling framework provides an exciting new avenue to elucidate the computational principles governing synaptic plasticity and learning in the brain.

Sighted animals use visual signals to discern directional motion in their environment. Motion is not directly detected by visual neurons, and it must instead be computed from light signals that vary over space and time. This makes visual motion estimation a near universal neural computation, and decades of research have revealed much about the algorithms and mechanisms that generate directional signals. The idea that sensory systems are optimized for performance in natural environments has deeply impacted this research. In this article, we review the many ways that optimization has been used to quantitatively model visual motion estimation and reveal its underlying principles. We emphasize that no single optimization theory has dominated the literature. Instead, researchers have adeptly incorporated different computational demands and biological constraints that are pertinent to the specific brain system and animal model under study. The successes and failures of the resulting optimization models have thereby provided insights into how computational demands and biological constraints together shape neural computation.