The current state of art in the literature indicates that linear visual receptive fields are Gaussian or formed based on Gaussian kernels in biological visual systems. In this paper, by employing hypotheses based on the anatomy and physiology of vertebrate biological vision, we propose a neural circuitry possessing Gaussian-related visual receptive fields. Here, we present a plausible circuitry system matching the characteristic properties of an ideal visual front end of biological visual systems and then present a condition under which this circuit demonstrates a linear behaviour to model the linear receptive fields observed in the biological experimental data. The objective of this study is to understand the hardware circuitry from which various visual receptive fields in biological visual system can be deduced. In our model, a nonlinear neural network communicating with spikes is considered. The condition under which this neural network behaves linearly is discussed. The equivalent linear circuit proposed here employs some anatomical and physiological properties of the early biological visual pathway to derive the visual receptive field profiles for linear cells such as neurons with isotropic separable, non-isotropic separable and non-separable (velocity-adapted) Gaussian receptive fields in the LGN and striate cortex. In the model presented here, the theory of transmission lines for linear distributed electrical circuits is employed for two-dimensional transmission grids to model cell connectivities in a neural layer. The model presented here leads to a formulation similar to the Gaussian scale-space theory for the transmission of visual signals through various layers of neurons. Our model therefore presents a new insight on how the convolution process with Gaussian kernels can be implemented in vertebrate visual systems. The comparison of the numerical simulations of our model presented in this paper with the data analysis of receptive field profiles recorded in the biological literature demonstrates a complete agreement between our theoretical model and experimental data. Our model is also in good agreement with the numerical results of the Gaussian scale-space theory for the visual receptive fields.