The PET-enabled dual-energy CT method allows dual-energy CT imaging on PET/CT scanners without the need for a second x-ray CT scan. A 511 keV γ-ray attenuation image can be reconstructed from time-of-flight PET emission data using the maximum-likelihood attenuation and activity (MLAA) algorithm. However, the attenuation image reconstructed by standard MLAA is commonly noisy. To suppress noise, we propose a neuralnetwork approach for MLAA reconstruction. The PET attenuation image is described as a function of kernelized neural networks with the flexibility of incorporating the available x-ray CT image as anatomical prior. By applying optimization transfer, the complex optimization problem of the proposed neural MLAA reconstruction is solved by a modularized iterative algorithm with each iteration decomposed into three steps: PET activity image update, attenuation image update, and neural-network learning using a weighed mean squared-error loss. The optimization algorithm is guaranteed to monotonically increase the data likelihood. The results from computer simulations demonstrated the neural MLAA algorithm can achieve a significant improvement on the γ-ray CT image quality as compared to other algorithms.