Theoretical analysis of neuronal network's response under different stimulus.

Neuromodulation plays a critical role in the normal physiological functions of organisms. With advancements in science and technology, neuromodulation has expanded into various fields. For instance, in the field of engineering, in vitro-cultured neural networks are utilized to perform closed-loop control for achieving complex functionalities. Conducting pioneering theoretical research using mathematical models is particularly essential for enhancing efficiency and reducing costs. This study focuses on examining the relationship between input and output in order to establish a groundwork for more advanced closed-loop regulation applications in engineering. Using a constructed neural network model, Poisson, square wave and direct current (DC) stimulation are applied. The results show that the network's firing rate increases with the frequency or amplitude of these stimulations. And the network's firing rate could reach to a stable state after the stimulation is applied for 0.8s and return to initial states when the stimulus is removed for 1s. To ascertain if the system exhibits a memory effect from the previous stimulus, we conduct independent and continuous stimulation schemes. Comparing the firing rate of neuronal networks under these two stimulation schemes reveals a memory effect of the system on the previous stimulus, which is independent of network properties and stimulus types. Finally, by applying square wave stimulation to the in vitro cultured neural network, we have confirmed that cultured neural network actually can reach to a steady state and have memory effects on the previous stimulus. Our research results have important theoretical significance and reference value for designing the closed-loop regulation strategy of in vitro cultured neuronal networks.