<p>Optical computing could reduce the energy cost of artificial intelligence by leveraging the parallelism and propagation speed of light. However, implementing nonlinear activation, essential for machine learning, remains challenging in low-power optical systems dominated by linear wave physics. Here, we introduce an optical spiking neural network that uses optical rogue-wave statistics to define a programmable firing mechanism. By establishing a homomorphism between free-space diffraction and neuronal integration, we demonstrate that phase-engineered caustics provide optimal states for extreme-event thresholding: sparse spatial spikes emerge when the local optical intensity is thresholded against a caustics-based rogue-wave criterion. Using a physics-informed digital twin, we optimize granular phase masks to deterministically concentrate energy into targeted detector regions, enabling end-to-end co-design of the optical transformation and a lightweight electronic readout. We experimentally validate the approach on BreastMNIST and Olivetti Faces, achieving accuracies of 82.45% and 96.67%, respectively, competitive with standard digital baselines. These results demonstrate that extreme-wave phenomena, often treated as deleterious fluctuations, can be harnessed as structural nonlinearity for scalable, energy-efficient neuromorphic photonic inference.</p>