High Dynamic Range (HDR) imaging with modulo cameras involves solving a challenging inverse problem, where degradation occurs due to the modulo operation applied to the target HDR image. Existing methods operate directly in the image domain, overlooking the underlying properties of the modulo operation. Motivated by Itoh's continuity condition in optics, we reformulate modulo HDR reconstruction in image gradient domain, leveraging the inherent properties of modulo-wrapped gradients to simplify the problem. Furthermore, to address possible ambiguities on large image gradients, we introduce an auxiliary variable with a learnable sparsity prior in an optimization formulation to absorb the related residuals. This is implemented within an unfolding network, where sparsity is enforced through a spiking neuron-based module. Experiments show that our method outperforms existing approaches while being among the lightest models of existing works.