Spiking neural network (SNN) is studied in multidisciplinary domains to (i) enable order-of-magnitudes energy-efficient AI inference and (ii) computationally simulate neuro-scientific mechanisms. The lack of discrete theory obstructs the practical application of SNN by limiting its performance and nonlinearity support. We present a new optimization-theoretic perspective of the discrete dynamics of spiking neurons. We prove that a discrete dynamical system of simple integrate-and-fire models approximates the sub-gradient method over unconstrained optimization problems. We practically extend our theory to introduce a novel sign gradient descent (signGD)-based neuronal dynamics that can (i) approximate diverse nonlinearities beyond ReLU and (ii) advance ANN-to-SNN conversion performance in low time steps. Experiments on large-scale datasets show that our technique achieves (i) state-of-the-art performance in ANN-to-SNN conversion and (ii) is the first to convert new DNN architectures, e.g., ConvNext, MLP-Mixer, and ResMLP. We publicly share our source code at https://github.com/snuhcs/snn_signgd .