The buildup of a cortical neuron's excitation, called the postsynaptic potential (PSP), is well modeled by the generalized inverse Gaussian (First) Hitting Time (GIGHT) diffusion. Such a model is called the generalized inverse Gaussian (GIG) neuron model. It is also believed that a neuron's purpose is to send information about the state of its input to the neuron's targets. We use Shannon's mutual information as a measure of this neural information in the GIG neuron model. The remarkable energy efficiency of neurons suggests that the neuron maximizes the transmitted mutual information subject to an energy constraint. Thus, we are interested in the information capacity of the GIG neuron model as a function of energy expended. This paper improves our work with Jie Xing [1], wherein the upper bound of the capacity-cost curve for the GIG neuron model was obtained. Here, we numerically computed the capacity-cost curve for such a model and found that for certain parameters, the upper bound and actual curve almost coincide. More interestingly, we found that the maximizing input distribution is discrete for certain parameters. This has implications on how the neural network should operate in order to optimally transmit information.