How brain circuits are organized to skillfully produce learned sequences of behaviors is still poorly understood. Here, we functionally examine how the 'cortical' song premotor region HVC, which is necessary for zebra finch song, controls the sequential production of learned song syllables. We find that HVC can generate the complete sequence of learned song syllables independently of its main synaptic input pathways. Thalamic input to HVC is permissive for song initiation but it is not required for transitions between syllables or completing song. We show that excitation of HVC neurons during song reliably causes vocalizations to skip back to the beginning of song, reminiscent of a skipping record. This restarting of syllable sequences can be induced at any moment in song and depends on local circuits within HVC. We identify and computationally model a synaptic network including intratelencephalic premotor and corticostriatal neurons within HVC that are essential for completing song syllable sequences. Together, our results show that the learned zebra finch song is controlled by a 'cortical' sequence-generating network in HVC that, once started, can sustain production of all song syllables independent of major extrinsic input pathways. Thus, sequential neuronal activity can be organized to fuse well-learned vocal-motor sequences, ultimately achieving holistic control of this naturally learned behavior.

Extensive studies suggest the brain performs Bayesian inference to infer the latent world states. It is a fundamental neuroscience question that how canonical recurrent neural circuits in the brain implement Bayesian inference. Many existing theoretical studies focused on how the recurrent circuits compute the posterior, while largely overlooking how the circuits compute the model evidence (normalization constant in Bayes' theorem), a key quantity that measures how well a model explains the observed data.Thus, it remains largely unknown about how the recurrent circuits compute the model evidence. The present study performs rigorous theoretical analyses of the continuous attractor networks, a canonical recurrent circuit model, and reveals that the nonlinear circuit dynamics can simultaneously compute the posterior and model evidence in first two dominant subspaces within the circuit dynamics. Specifically, the circuit dynamics in the stimulus feature subspace implements the Langevin posterior sampling, and the circuit dynamics in the subspace of total neuronal activity computes the model evidence in a way analogous to the evidence lower bound in stochastic variational inference. We further extend the circuit model to compute the model evidence of multiple inputs, and simulations validate the computation in the network. Our work for the first time reveals the concurrent model evidence and posterior sampling in subspaces in continuous attractor networks, significantly deepen our understanding of the computational algorithms adopted by the neural circuits.

Zero-shot generalization to unseen embodiments is important for generalizable vision-language-action (VLA) models as robot hardware evolves and task-specific data collection remains costly. However, a systematic understanding of this problem remains limited, in part because the literature lacks a unified zero-shot transfer definition and controlled evaluation settings that isolate embodiment changes from differences in tasks, scenes, or protocols. To address this gap, we first distinguish strict zero-shot transfer, where the target embodiment is absent from all training data, from pretrain-exposed zero-shot transfer, where it appears only during pretraining. We then introduce a controlled benchmark spanning 14 held-out target embodiments across simulation and real-world validation. Within this framework, we conduct a controlled analysis of four factors: state-action representations, pretraining embodiment diversity, auxiliary co-training objectives, and target-embodiment exposure. Experimental results show that local end-effector (EEF) state-action representations, the source embodiment diversity, and auxiliary co-training improve cross-embodiment transfer by around 15, 18, and 7 percentage points, respectively. We further find that adding only 5

Navigation in complex environments relies on internal spatial representations that guide action. While the brain employs a diverse repertoire of spatial tuning cells—including grid, place, and head-direction cells—a normative theory linking these static neural codes to the dynamic process of navigation remains elusive. In this work, we propose a Unified Laplacian Framework derived from first principles of representational smoothness and efficiency. We first demonstrate that diverse spatial codes emerge naturally as spectral decompositions of the Laplacian operator. Crucially, bridging the gap from representation to action, we derive a biologically plausible navigation policy based on the Green's function potential. We show that this potential encodes the environment's intrinsic geometry to enable simple, trap-free gradient ascent, achieving significantly improved sample efficiency and generalization in goal-reaching tasks. Furthermore, we demonstrate that these spectral representations can be learned directly from high-dimensional visual inputs, confirming its plausibility in realistic environments. Our results suggest that the "cognitive map" can be viewed as a spectral embedding of the Laplacian, providing a rigorous foundation for spatial cognition in both biological and artificial agents.

Cognitive reasoning, also known as mental programming, is fundamental to our intelligence. A central function in mental programming is working memory (WM), involving both temporarily maintaining the information and manipulating it based on rules. While the neural mechanisms of WM maintenance have been extensively studied, those governing WM manipulation remain largely unknown. To bridge this gap, the present study focuses on an elementary operation in WM manipulation: re-ordering items in WM sequences. We propose a functional, biologically plausible neural circuit model that consists of two interconnected modules: a memory module composed of continuous attractor-based memory slots that store item features, and a control module sending gain-modulating commands to orchestrate specific operations in the memory module. The model successfully implements two-item swapping in a WM sequence, generating neuronal responses similar to recent primate experiments of WM sequence manipulation. By incorporating principles from the algebraic permutation group, we generalize the circuit model to accommodate more complex sequence manipulations. This math foundation reveals how arbitrary permutations can be decomposed into sequences of elementary swapping operations, which can be generated by a hierarchical tree-structured control circuit module. And the mutual inhibition within the control tree ensures that only one program is being executed at the same time. Our study establishes overarching connections among mental programming neural circuit models, neuro-science experiments, and abstract algebraic structure. These insights enhance our understanding of the neural underpinnings of cognitive reasoning and inspire the design of artificial cognitive systems. ### Competing Interest Statement The authors have declared no competing interest. The University of Texas Southwestern Medical Center, https://ror.org/05byvp690, UT Southwestern Medical Center Endowed Scholars program Science Fund for Creative Research Groups of the National Natural Science Foundation of China, T2421004 National Key Research and Development Program of China, 2024YFF1206500

The brain lives in an ever-changing world and needs to infer the dynamic evolution of latent states from noisy sensory inputs. Exploring how canonical recurrent neural circuits in the brain realize dynamic inference is a fundamental question in neuroscience. Nearly all existing studies on dynamic inference focus on deterministic algorithms, whereas cortical circuits are intrinsically stochastic, with accumulating evidence suggesting that they employ stochastic Bayesian sampling algorithms. Nevertheless, nearly all circuit sampling studies focused on static inference with fixed posterior over time instead of dynamic inference, leaving a gap between circuit sampling and dynamic inference. To bridge this gap, we study the sampling-based dynamic inference in a canonical recurrent circuit model with excitatory (E) neurons and two types of inhibitory interneurons: parvalbumin (PV) and somatostatin (SOM) neurons. We find that the canonical circuit unifies Langevin and Hamiltonian sampling to infer either static or dynamic latent states with various moving speeds. Remarkably, switching sampling algorithms and adjusting model’s internal latent moving speed can be realized by modulating the gain of SOM neurons without changing synaptic weights. Moreover, when the circuit employs Hamiltonian sampling, its sampling trajectories oscillate around the true latent moving state, resembling the decoded spatial trajectories from hippocampal theta sequences. Our work provides overarching connections between the canonical circuit with diverse interneurons and sampling-based dynamic inference, deepening our understanding of the circuit implementation of Bayesian sampling. ### Competing Interest Statement The authors have declared no competing interest.

We present a minimalistic representation model for the head direction (HD) system, aiming to learn a high-dimensional representation of head direction that captures essential properties of HD cells. Our model is a representation of rotation group U(1), and we study both the fully connected version and convolutional version. We demonstrate the emergence of Gaussian-like tuning profiles and a 2D circle geometry in both versions of the model. We also demonstrate that the learned model is capable of accurate path integration.

Accumulating evidence suggests the canonical cortical circuit, consisting of excitatory (E) and diverse classes of inhibitory (I) interneurons, implement sampling-based Bayesian inference to compute stimulus posteriors. However, most of the identified sampling algorithms in the circuit are still simpler than the nonlinear circuit dynamics. Through comprehensive theoretical analyses of the nonlinear circuit dynamics, we discover the canonical circuit dynamics innately implements natural gradient Bayesian sampling, which is an advanced sampling algorithm that adaptively adjusts the sampling step size based on the local geometry of stimulus posteriors measured by Fisher information. Specifically, the nonlinear circuit dynamics can implement natural gradient Langevin and Hamiltonian sampling of uniand multi-variate stimulus posteriors, and these algorithms can be switched by interneurons. We also find that the non-equilibrium circuit dynamics when transitioning from the resting to evoked state can further accelerate natural gradient sampling, and analytically identify the neural circuit’s annealing strategy. Remarkably, we identify the approximated computational strategies employed in the circuit dynamics, which even resemble the ones widely used in machine learning. Our work provides an overarching connection between canonical circuit dynamics and advanced sampling algorithms, deepening our understanding of the circuit algorithms of Bayesian sampling. ### Competing Interest Statement The authors have declared no competing interest. UT Southwestern Endowed Scholars program

This paper investigates the conformal isometry hypothesis as a potential explanation for the hexagonal periodic patterns in grid cell response maps. We posit that grid cell activities form a high-dimensional vector in neural space, encoding the agent's position in 2D physical space. As the agent moves, this vector rotates within a 2D manifold in the neural space, driven by a recurrent neural network. The conformal hypothesis proposes that this neural manifold is a conformal isometric embedding of 2D physical space, where local physical distance is preserved by the embedding up to a scaling factor (or unit of metric). Such distance-preserving position embedding is indispensable for path planning in navigation, especially planning local straight path segments. We conduct numerical experiments to show that this hypothesis leads to the hexagonal grid firing patterns by learning maximally distance-preserving position embedding, agnostic to the choice of the recurrent neural network. Furthermore, we present a theoretical explanation of why hexagon periodic patterns emerge by minimizing our loss function by showing that hexagon flat torus is maximally distance preserving.

The hippocampus supports spatial navigation by encoding cognitive maps through collective place cell activity. We model the place cell population as non-negative spatial embeddings derived from the spectral decomposition of multi-step random walk transition kernels. In this framework, inner product or equivalently Euclidean distance between embeddings encode similarity between locations in terms of their transition probability across multiple scales, forming a cognitive map of adjacency. The combination of non-negativity and inner-product structure naturally induces sparsity, providing a principled explanation for the localized firing fields of place cells without imposing explicit constraints. The temporal parameter that defines the diffusion scale also determines field size, aligning with the hippocampal dorsoventral hierarchy. Our approach constructs global representations efficiently through recursive composition of local transitions, enabling smooth, trap-free navigation and preplay-like trajectory generation. Moreover, theta phase arises intrinsically as the angular relation between embeddings, linking spatial and temporal coding within a single representational geometry.

Grid cells in the entorhinal cortex of the mammalian brain exhibit striking hexagon firing patterns in their response maps as the animal (e.g., a rat) navigates in a 2D open environment. The responses of the population of grid cells collectively form a vector in a high-dimensional neural activity space, and this vector represents the self-position of the agent in the 2D physical space. As the agent moves, the vector is transformed by a recurrent neural network that takes the velocity of the agent as input. In this paper, we propose a simple and general conformal normalization of the input velocity for the recurrent neural network, so that the local displacement of the position vector in the high-dimensional neural space is proportional to the local displacement of the agent in the 2D physical space, regardless of the direction of the input velocity. Our numerical experiments on the minimally simple linear and non-linear recurrent networks show that conformal normalization leads to the emergence of the hexagon grid patterns. Furthermore, we derive a new theoretical understanding that connects conformal normalization to the emergence of hexagon grid patterns in navigation tasks.

Accumulating evidence suggests stochastic cortical circuits can perform sampling-based Bayesian inference to compute the latent stimulus posterior. Canonical cortical circuits consist of excitatory (E) neurons and types of inhibitory (I) interneurons. Nevertheless, nearly no sampling neural circuit models consider the diversity of interneurons, and thus how interneurons contribute to sampling remains poorly understood. To provide theoretical insight, we build a nonlinear canonical circuit model consisting of recurrently connected E neurons and two types of I neurons including Parvalbumin (PV) and Somatostatin (SOM) neurons. The E neurons are modeled as a canonical ring (attractor) model, receiving global inhibition from PV neurons, and locally tuning-dependent inhibition from SOM neurons. We theoretically analyze the nonlinear circuit dynamics and analytically identify the Bayesian sampling algorithm performed by the circuit dynamics. We found a reduced circuit with only E and PV neurons performs Langevin sampling, and the inclusion of SOM neurons with tuning-dependent inhibition speeds up the sampling via upgrading the Langevin into Hamiltonian sampling. Moreover, the Hamiltonian framework requires SOM neurons to receive no direct feedforward connections, consistent with neuroanatomy. Our work provides overarching connections between nonlinear circuits with various types of interneurons and sampling algorithms, deepening our understanding of circuit implementation of Bayesian inference.

The information processing in the brain and embodied agents form a sensory-action loop to interact with the world. An important step in the loop is motion planning which selects motor actions based on the current world state and task need. In goal-directed navigation, the brain chooses and generates motor actions to bring the current state into the goal state. It is unclear about the neural circuit mechanism of motor action selection, nor its underlying theory. The present study formulates the motion planning as a Lie group operator search problem, and uses the 1D rotation group as an example to provide insight into general operator search in neural circuits. We found the abstract group operator search can be implemented by a two-layer feedforward circuit utilizing circuit motifs of connection phase shift, nonlinear activation function, and pooling, similar to Drosophila's goal-directed navigation neural circuits. And the computational complexity of the feedforward circuit can be even lower than common signal processing algorithms in certain conditions. We also provide geometric interpretations of circuit computation in the group representation space. The feedforward motion planning circuit is further combined with sensory and motor circuit modules into a full circuit of the sensory-action loop implementing goal-directed navigation. Our work for the first time links the abstract operator search with biological neural circuits.

Grid cells in the entorhinal cortex of mammalian brains exhibit striking hexagon grid firing patterns in their response maps as the animal (e.g., a rat) navigates in a 2D open environment. In this paper, we study the emergence of the hexagon grid patterns of grid cells based on a general recurrent neural network (RNN) model that captures the navigation process. The responses of grid cells collectively form a high dimensional vector, representing the 2D self-position of the agent. As the agent moves, the vector is transformed by an RNN that takes the velocity of the agent as input. We propose a simple yet general conformal normalization of the input velocity of the RNN, so that the local displacement of the position vector in the high-dimensional neural space is proportional to the local displacement of the agent in the 2D physical space, regardless of the direction of the input velocity. We apply this mechanism to both a linear RNN and nonlinear RNNs. Theoretically, we provide an understanding that explains the connection between conformal normalization and the emergence of hexagon grid patterns. Empirically, we conduct extensive experiments to verify that conformal normalization is crucial for the emergence of hexagon grid patterns, across various types of RNNs. The learned patterns share similar profiles to biological grid cells, and the topological properties of the patterns also align with our theoretical understanding.

Time perception is fundamental in our daily life. An important feature of time perception is temporal scaling (TS): the ability to generate temporal sequences (e.g., movements) with different speeds. However, it is largely unknown about the mathematical principle underlying TS in the brain. The present theoretical study investigates temporal scaling from the Lie group point of view. We propose a canonical nonlinear recurrent circuit dynamics, modeled as a continuous attractor network, whose neuronal population responses embed a temporal sequence that is TS equivariant. We find the TS group operators can be explicitly represented by a time-invariant control input to the network, whereby the input gain determines the TS factor (group parameter), and the spatial offset between the control input and the network state on the continuous attractor manifold gives rise to the generator of the Lie group. The recurrent circuit’s neuronal responses are consistent with experimental data. The recurrent circuit can drive a feedforward circuit to generate complex sequences with different temporal scales, even in the case of negative temporal scaling (“time reversal”). Our work for the first time analytically links the abstract temporal scaling group and concrete neural circuit dynamics.

Experimental data on multisensory integration revealed that modules in the brain processing different inputs are reciprocally connected, and can be utilized for multisensory information integration.We consider environments described by composite priors with both correlated and independent components, and reveal that the correlated and composite posterior probabilities can be encoded in the collective neural activities of separate neuron groups, referred to as congruent and integrated groups, with the latter receiving feedforward inputs from the former.Furthermore, in the framework of probabilistic population coding, we find that the accuracy of the Bayesian prediction can be remarkably improved if the noise in the neuronal dynamics is determined by the amplitude of the population vector, which originates from the collective nature of the neuronal responses themselves.

Equivariant representation is necessary for the brain and artificial perceptual systems to faithfully represent the stimulus under some (Lie) group transformations. However, it remains unknown how recurrent neural circuits in the brain represent the stimulus equivariantly, nor the neural representation of abstract group operators. The present study uses a one-dimensional (1D) translation group as an example to explore the general recurrent neural circuit mechanism of the equivariant stimulus representation. We found that a continuous attractor network (CAN), a canonical neural circuit model, self-consistently generates a continuous family of stationary population responses (attractors) that represents the stimulus equivariantly. Inspired by the Drosophila's compass circuit, we found that the 1D translation operators can be represented by extra speed neurons besides the CAN, where speed neurons' responses represent the moving speed (1D translation group parameter), and their feedback connections to the CAN represent the translation generator (Lie algebra). We demonstrated that the network responses are consistent with experimental data. Our model for the first time demonstrates how recurrent neural circuitry in the brain achieves equivariant stimulus representation.

The activity of the grid cell population in the medial entorhinal cortex (MEC) of the mammalian brain forms a vector representation of the self-position of the animal. Recurrent neural networks have been proposed to explain the properties of the grid cells by updating the neural activity vector based on the velocity input of the animal. In doing so, the grid cell system effectively performs path integration. In this paper, we investigate the algebraic, geometric, and topological properties of grid cells using recurrent network models. Algebraically, we study the Lie group and Lie algebra of the recurrent transformation as a representation of self-motion. Geometrically, we study the conformal isometry of the Lie group representation where the local displacement of the activity vector in the neural space is proportional to the local displacement of the agent in the 2D physical space. Topologically, the compact abelian Lie group representation automatically leads to the torus topology commonly assumed and observed in neuroscience. We then focus on a simple non-linear recurrent model that underlies the continuous attractor neural networks of grid cells. Our numerical experiments show that conformal isometry leads to hexagon periodic patterns in the grid cell responses and our model is capable of accurate path integration. Code is available at \url{https://github.com/DehongXu/grid-cell-rnn}.

Hippocampal place cells of freely moving rodents display an intriguing temporal organization in their responses known as `theta phase precession', in which individual neurons fire at progressively earlier phases in successive theta cycles as the animal traverses the place fields. Recent experimental studies found that in addition to phase precession, many place cells also exhibit accompanied phase procession, but the underlying neural mechanism remains unclear. Here, we propose a neural circuit model to elucidate the generation of both kinds of phase shift in place cells' firing. Specifically, we consider a continuous attractor neural network (CANN) with feedback inhibition, which is inspired by the reciprocal interaction between the hippocampus and the medial septum. The feedback inhibition induces intrinsic mobility of the CANN which competes with the extrinsic mobility arising from the external drive. Their interplay generates an oscillatory tracking state, that is, the network bump state (resembling the decoded virtual position of the animal) sweeps back and forth around the external moving input (resembling the physical position of the animal). We show that this oscillatory tracking naturally explains the forward and backward sweeps of the decoded position during the animal's locomotion. At the single neuron level, the forward and backward sweeps account for, respectively, theta phase precession and procession. Furthermore, by tuning the feedback inhibition strength, we also explain the emergence of bimodal cells and unimodal cells, with the former having co-existed phase precession and procession, and the latter having only significant phase precession. We hope that this study facilitates our understanding of hippocampal temporal coding and lays foundation for unveiling their computational functions.

Recent advances in deep generative models have led to immense progress in 3D shape synthesis. While existing models are able to synthesize shapes represented as voxels, point-clouds, or implicit functions, these methods only indirectly enforce the plausibility of the final 3D shape surface. Here we present a 3D shape synthesis framework (SurfGen) that directly applies adversarial training to the object surface. Our approach uses a differentiable spherical projection layer to capture and represent the explicit zero isosurface of an implicit 3D generator as functions defined on the unit sphere. By processing the spherical representation of 3D object surfaces with a spherical CNN in an adversarial setting, our generator can better learn the statistics of natural shape surfaces. We evaluate our model on large-scale shape datasets, and demonstrate that the end-to-end trained model is capable of generating high fidelity 3D shapes with diverse topology.