In this paper, a generic model of memristive systems, which can emulate the behavior of real memristive devices is proposed. Non-ideal pinched hysteresis loops are sometimes observed in real memristive devices. For example, the hysteresis loops may deviate from the origin over a broad range of amplitude A and frequency f of the input signal. This deviation from the ideal case is often caused by parasitic circuit elements exhibited by real memristive devices. In this paper, we propose a generic memristive circuit model by adding four parasitic circuit elements, namely, a small capacitance, a small inductance, a small DC current source, and a small DC voltage source, to the memristive device. The adequacy of this model is verified experimentally and numerically with two thermistors (NTC and PTC) memristors.

This paper presents an in-depth analysis of an excitable membrane of a biological system by proposing a novel approach that the cells of the excitable membrane can be modeled as the networks of memristors. We provide compelling evidence from the Chay neuron model that the state-independent mixed ion channel is a nonlinear resistor, while the state-dependent voltage-sensitive potassium ion channel and calcium-sensitive potassium ion channel function as generic memristors from the perspective of electrical circuit theory. The mechanisms that give rise to periodic oscillation, aperiodic (chaotic) oscillation, spikes, and bursting in an excitable cell are also analyzed via a small-signal model, a pole-zero diagram of admittance functions, local activity, the edge of chaos, and the Hopf bifurcation theorem. It is also proved that the zeros of the admittance functions are equivalent to the eigen values of the Jacobian matrix, and the presence of the positive real parts of the eigen values between the two bifurcation points lead to the generation of complicated electrical signals in an excitable membrane. The innovative concepts outlined in this paper pave the way for a deeper understanding of the dynamic behavior of excitable cells, offering potent tools for simulating and exploring the fundamental characteristics of biological neurons.

This paper investigates the essential role of local activity theorem in pancreatic $\beta$ -cells in the process of insulin secretion. It is shown that the ion channels in $\beta$ -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 $\beta$ -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 $\beta$ -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 $\beta$ -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 $\beta$ -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.