Fig. 3: Energy-representational manifold and its characteristics.

A Left: Schematic illustration of the object manifold in high-dimensional neural space. Face stimuli are located on the red line segment, while tool stimuli are positioned on the black line segment. Objects that share shapes or configurations with faces are situated in between. The images shown here were generated by artificial intelligence for illustration purpose only. Middle: The 2D representational manifold of object categories projected from the high-dimensional neural space via the parametric-UMAP method. Each small dot represents one object category at the basic level (e.g., cat face), and colors of the dots indicate object categories at the conceptual level (e.g., faces of different species). Big dots denote the center of object categories at the conceptual level. Right: Energy of neural states incorporated into the 2D representational manifold as a third axis, thus forming a 3D energy-representation manifold. Contour lines projected onto the representational manifold denote energy values, illustrating the shape and elevation of the energy. Contour lines close together indicate a steep slope, while lines spaced further apart indicate a gentler slope. B Left: Energy-representation manifolds constructed by models with a longer (\(\lambda\) = 0.0, left) and shorter (\(\lambda\) = 0.1, right) wiring length, respectively. Note that contour lines with a longer wiring length (top) are closer together than those with a shorter wiring length (bottom), indicating a steeper slope in the changes of energy in the manifold. Middle: Schematic illustration of the attractor regions in the energy-representation manifolds with different slopes. The steeper the slope, the longer the path that takes the initial neural state to travel to the stable state. Right: Average path lengths of attractor regions in models with different wiring lengths. Our model with the biologically inspired wiring length formed attractor regions that were neither too deep nor too shallow, suggesting a balanced dynamic. C Criticality. Hamiltonian value (top) and specific heat (bottom) at different temperatures of the Ising model. The Hamiltonian value for the model with \(\lambda\) = 0.023 lay close to the critical point of the Ising model, whereas models with either a longer or a shorter wiring length resided in either order or chaos regions. D Cognitive impenetrability. Left: A top-down signal of faces was applied to all neurons (indicative of the presence of faces), while inputs were continuously presented objects (e.g., lemon, ambulance, and backpack). Right: A top-down signal of objects was applied to all neurons (indicative of the presence of objects), while input stimuli were continuously presented various faces. Lines denote time courses of activation averaged across all neurons in the face cluster along iteration steps, with shaded areas denoting standard deviation. Time courses show the dominant effect of top-down signals (duration: from 80,000 to 160,000 steps); however, once the top-down signals were offset, the time courses resumed to their original trajectories, suggesting the cognitive impenetrability of the model. ***p < 0.001.