The Izhikevich neuron model is a widely adopted computational neuron model that comprises a set of quadratic differential equations involving two variables. Consequently, obtaining a closed-form solution is unattainable, making it challenging to perform further rate-coding analysis. In this study, we establish a balanced background noise Izhikevich neuron model with periodic signal input. Treating the system of differential equations as a velocity vector field, we are able to compute the Hamiltonian energy function for this model. The interspike-interval firing rate function is then derived with the aid of the Hamiltonian function. Using the firing rate function, we propose a solution curve on the novel \(\gamma \) - \(\gamma '\) phase plane for better understanding the timing when extreme values of the firing rate function occur. Additionally, we address a phase advance phenomenon that occurs between the sinusoidal current injection and the interspike-interval firing rate curve, attempting to provide a qualitative explanation for this phenomenon.