Neural Receptive Fields, Stimulus Space Embedding and Effective Geometry of Scale-Free Networks

TL;DR

The paper addresses how receptive fields and population-level dynamics align with external stimulus spaces in brain networks without ad hoc synaptic tuning. It proposes a hyperbolic-geometry framework for scale-free networks, embedding nodes in $\mathbb{H}^d$ and associating the stimulus space with the boundary so that RFs and bump attractors arise from geometry, with $S \propto e^{-\alpha r}$ and $S \propto k^{-\beta}$. They validate the approach with rate-based and spiking simulations and demonstrate that RF sizes are exponentially distributed and dependent on embedding radius, replicated in hippocampal place-field data from rats on a linear track, including curvature peaks at track junctions. The work provides a unifying principle linking network topology, stimulus encoding, and neural dynamics across modalities and dimensionalities, with predictions such as RF size scaling with neuron degree and robustness across varied conditions.

Abstract

Understanding how receptive fields emerge and organize within brain networks and how neural dynamics couple with stimuli space is fundamental to neuroscience. Models often rely on fine-tuning connectivity to match empirical data, which may limit biological plausibility. Here we propose a physiologically grounded alternative where receptive fields and population-level attractor dynamics arise naturally from the effective hyperbolic geometry of scale-free networks. By associating stimulus space with the boundary of a hyperbolic embedding, we simulate neural dynamics using rate-based and spiking models, revealing localized activity patterns that reflect stimulus space structure without synaptic fine-tuning. The resulting receptive fields follow experimentally observed statistics and properties, and their sizes depends on neuron's connectivity degree. The model generalizes across stimuli dimensionalities and various modalities, such as orientation and place selectivity. Experimental analyses of hippocampal place fields recorded on a linear track support these findings. This framework offers a novel organizing principle linking network structure, stimulus space encoding, and neural dynamics, providing insights into receptive field formation across diverse brain areas.

Figures (9)

• Figure 1: Neural responses are governed by the receptive fields.(a) A map of angular domains---receptive fields of head-direction cells---covering configurational space of orientations, i.e., a topological cycle. (Blue dots schematically represent the firing of individual head-direction neuron, similar to the place cell responses illustrated in panel b.) (b) Hippocampal place cells spike when the animal appears in particular regions in the navigated environment (red circular domain). Place fields tile the navigated environment and spiking is localized on neurons whose receptive fields intersect with the animal's current position in space, thus reflecting the current stimulus state. (c) The proposed approach enables a principled coupling between stimulus spaces and network dynamics without compromising structural flexibility. The stimulus space can be associated with an effective boundary space of the network’s hyperbolic embedding due to the geometric properties of scale-free networks. Receptive-field organization inherits this hyperbolic geometry, and the resulting family of localized activity states mirrors the structure of the stimulus space.
• Figure 2: Hyperbolic spaces and scale-free networks.(a) Three isotropic $2D$ geometries: flat Euclidean space (left), spherical space with positive curvature (right), and hyperbolic space with negative curvature (bottom). (b) Poincaré model: the entire $2D$ Lobachevsky plane, $\mathbb{H}^2$, is mapped conformally onto the compact Euclidean disk. The boundary of the disk, corresponding to $S^1$, represents points at infinity, enclosing $\mathbb{H}^2$. Geodesic segments (blue lines) form angle-deficient triangles that tile the entire $\mathbb{H}^2$ space. (c) Scale-free networks exhibit an effective hyperbolic structure that allows embedding them within the Poincaré disk model. Each node $v$ is assigned a radial coordinate (embedding radius) reflecting its degree---nodes with higher connectivity (hubs) are positioned closer to the center, while weakly connected nodes are pushed toward the boundary. Smaller angular distance between nodes reflects similarity in local connectivity structure and a higher probability of being connected.
• Figure 3: Localized attractors over the stimulus spaces. (a) Hyperbolic network embedding into the Poincaré disk. The radial coordinate is associated with node degree. Color indicates the density of a neuron's neighbors along the boundary in a characteristic excitation profile. If the input distributes over the entire boundary, the activity shifts from the periphery to the bulk. (b) The effective negative curvature reflects the power-law distribution of node degrees. (c) The excitatory response to a stimulus field on the boundary (black dots). Response activity is localized on a small subset of neurons and suppressed on the periphery, induced solely by the network's effective geometry, without synaptic fine-tuning. (d) Network activity induced by stimuli of different widths and positions over $\sim$1.5 seconds. The amplitude of the bump attractor remains stable across stimulus sizes and locations (red line shows peak firing frequency). (e) Distribution of response activity across a scale-free network (cf. panel c).
• Figure 4: Hyperbolicity of Receptive Fields and Scale Sensitivity.(a) The receptive fields of neurons in a hyperbolic network produce a complex tiling of the stimulus space. (b) The sizes of receptive fields in the frequency-based model correlate with the neurons' radial coordinates in the hyperbolic embedding. (c) The distribution of receptive field sizes follows an exponential law, consistent with experimental observations zhang2023hippocampal. (d) Localization of neural responses in a simulated hyperbolic network arises as a consequence of hyperbolicity---i.e., exponential concentration of cells toward the boundary. (e) A small stimulus displacement affects neurons with different receptive field scales differently: large receptive fields respond to broader shifts, while nested smaller fields exhibit greater sensitivity to fine changes.
• Figure 5: Place field scaling. Different widths and receptive field sizes in the firing-rate model. (a) Distribution of neuron activity across the stimulus space in the spiking model for both narrow and wide stimuli. (b) Distribution of receptive field sizes for a narrow stimulus. (c) Dependence of receptive field size on the neuron's radial coordinate for stimuli of different widths. (d) Shift in the distribution of receptive fields with increasing characteristic stimulus width. (e) Ratio of receptive field sizes between the spiking and rate-based models (with the narrowest stimulus width fixed in the rate model).