CTKAN: MAKING KOLMOGOROV–ARNOLD NETWORKS THINK OVER SPLINE ACTIVITIES FOR MEDICAL IMAGE SEGMENTATION

Neural computation can benefit from evolving internal dynamics rather than relying solely on a single static mapping. Although Kolmogorov–Arnold Networks (KANs) have shown promise in medical image segmentation through learnable spline functions, most KAN-based models evaluate spline responses once and aggregate them immediately, preventing basis-wise contributions from being reconsidered before the layer output is formed. We argue that KANs should think over their spline activities through iterative collaborative thinking, repeatedly refining and coordinating basis contributions before aggregation to strengthen nonlinear representation. We therefore propose CTKAN, a Continuous-Thought Kolmogorov–Arnold Network that introduces neural-dynamics-inspired recurrent evolution into the spline activity space. CTKAN maintains a sample-conditioned state for each output–input–basis triplet and updates these states over multiple thought ticks. Normalized pairwise co-activities provide relational feedback, while terminal states generate residual, sample-adaptive gates over shared spline coefficients, allowing each sample to induce a distinct mixture of spline bases. We integrate CTKAN at two deployment scales into a U-shaped segmentation backbone. Experiments on ultrasound, endoscopic, histopathological, and cardiac MRI benchmarks show that both variants outperform U-KAN, with CTKANMax improving the four-dataset macro-average by 1.29 IoU and 1.03 Dice points. These results demonstrate that iterative collaborative thinking over spline activities improves segmentation performance through sample-adaptive spline modulation.