This paper investigates the essential role of local activity theorem in pancreatic β-cells in the process of insulin secretion. It is shown that the ion channels in β-cells distributed over the pancreas are in fact generic memristors from the perspective of electrical circuit theory. Through our comprehensive analyses and extensive simulations from the Chay-Keizer Pancreatic β-model, Phantom Bursting Model (PBM)-I and PBM-II, this paper rigorously provides an ambient proof that the secretion and release of the insulin in the form of bursting or action potential are possible only when it satisfies the condition of local activity theorem. The local activity principle is analyzed by the small signal admittance function, pole zero diagram, and edge of chaos theorem. It is shown that the external stimulus or cell parameters chosen within the subset of the locally domain regime where the negative real part of the admittance function and the positive real part of the zeros (equivalent to the Eigen values) lead to the generation of complicated electrical signals in pancreatic β-cells. The examples presented in this paper demonstrate the local activity theorem is a novel and efficient tool for testing whether the secretion of insulin by pancreatic β-cells are possible or not. It follows from our in-depth analysis that the local activity is essential for the synthesis, secretion, and release of insulin by pancreatic β-cells. This principle serves as a critical condition, highlighting the intrinsic role of cellular dynamics in insulin regulation and emphasizing its significance in understanding pancreatic function and metabolic health.