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Research is conducted on the discharge rhythm properties of an artificial biological neural network with lateral inhibition mechanism, which is modeled by Hodgkin–Huxley (HH) equations and coupled by small-world (SW) network. The results show that the excitation frequency increases almost linearly along with increasing stimulation intensity, conversely, the average excitation intensity decreases near linearly with the increase of stimulation intensity, when the current stimulation intensity exceeds threshold value. In addition, the influence of coupling strength on excitation properties is noticeable under relatively weak stimulation. The stronger is the coupling strength, the lower is the excitation frequency, whereas the stronger is the coupling strength the greater is the excitation intensity. However, the influence of coupling strength on excitation properties is unnoticeable under relatively strong stimulation. For comparison, discharge rhythm properties of the neural networks under random connection and full connection are also investigated.

The four-dimensional Hodgkin–Huxley equations are considered as the prototype for description of neural pulse propagation. Their mathematical complexity and sophistication prompted a simplified two-dimensional model, the FitzHugh–Nagumo equations, which display many of the former's dynamical features. Numerical and mathematical analysis are employed to demonstrate that the FitzHugh–Nagumo equations can provide quantitative predictions in close agreement with the Hodgkin–Huxley equations. The two most important parameters of a neural pulse are its speed c(T) and pulse height v_max(T) and so numerical computations of these quantities predicted by the Hodgkin–Huxley equations are given over the entire temperature range T for stability of a neural pulse. Similarly, the FitzHugh–Nagumo equations are parameterized by two dimensionless quantities: a which determines the dynamics of the pulse front, and b whose departure from zero tailors the front to form the resultant pulse. Parallel computations are presented for the FitzHugh–Nagumo pulse whose relative simplicity permits analytic determination to close approximation of the dimensionless speed θ(a, b) and pulse height V_max(a, b). It is shown that the two models are numerically identified by scaling according to c = 4904 θ cm/sec and v_max = 115 V_maxmV where the numbers are a consequence of the experimental parameter values inherent to the Hodgkin–Huxley equations. With this connection, at a given temperature the Hodgkin–Huxley speed and pulse height determine unique values for the two FitzHugh–Nagumo parameters a and b. Approximate analytic solution for θ(a, b) allows construction of a three-dimensional [a, b, θ] state plot upon which a unique ridge defines, as a function of temperature, the speed and associated pulse height predicted by the Hodgkin–Huxley equations. The generality of the state plot suggests its application to other conductance models. Comparison of the Hodgkin–Huxley with the FitzHugh–Nagumo models highlight the quantitative limitations of the latter in the region of the minimum characterizing the back portion of the pulse. To overcome this limitation would require analytic extension of the FitzHugh–Nagumo dynamics to higher dimensionality.

The four-dimensional Hodgkin–Huxley equations describe the propagation in space and time of the action potential v(z) along a neural axon with z = x + ct and c being the pulse speed. The potential v(z), which is parameterized by the temperature, is driven by three gating functions, m(z), n(z) and h(z), each of which obeys formal first order kinetics with rate constants that are represented as nonlinear functions of the potential v. It is shown that this system can be analytically simplified (i) in the number of gating functions and (ii) in the form of associated rate functions while retaining to close approximation quantitative fidelity to computer solutions of the exact equations over the complete temperature range for which stable pulses exist. At a given temperature we record two solutions (T < T_max) corresponding to a high-speed and a low-speed branch in speed-temperature plots, c(T), or no solution (T > T_max). The pulse is considered as composed of two contiguous parts: (i) a pulse front extending from v(0) = 0 to a pulse maximum v = V_max, and (ii) a pulse back extending from V_max through a pulse minimum V_min to a final regression back to v(z → ∞) = 0. An approximate analytic solution is derived for the pulse front, which is predicted to propagate at a speed c(T) = 1203 Θ^⅜ (T°C) cm/sec, in close agreement with computer solution of the exact Hodgkin–Huxley equations for the entire pulse. These results provide the basis for a derivation of two-dimensional differential equation systems for the pulse front and pulse back, which predict the pulse maximum and minimum over the operational temperature range 0 ≤ T ≤ 25°C, in close agreement with the exact equations. Most neuron dynamics studies have been based on voltage clamp experiments featuring external current injection in place of self-generating pulse propagation. Since the behaviors of the gating functions are similar, it is suggested that the present approximations might be applicable to such situations as well as to the dynamics of myelinated fibers.

This paper presents a rigorous and comprehensive nonlinear circuit-theoretic foundation for the memristive Hodgkin–Huxley Axon Circuit model. We show that the Hodgkin–Huxley Axon comprises a potassium ion-channel memristor and a sodium ion-channel memristor, along with some mundane circuit elements. From this new perspective, many hitherto unresolved anomalous phenomena and paradoxes reported in the literature are explained and clarified. The yet unknown nonlinear dynamical mechanisms which give birth to the action potentials remain hidden within the memristors, and the race is on for uncovering the ultimate truth.

This paper shows the action potential (spikes) generated from the Hodgkin–Huxley equations emerges near the edge of chaos consisting of a tiny subset of the locally active regime of the HH equations. The main result proves that the eigenvalues of the 4 × 4 Jacobian matrix associated with the mathematically intractable system of four nonlinear differential equations are identical to the zeros of a scalar complexity function from complexity theory. Moreover, we show the loci of a pair of complex-conjugate zeros migrate continuously as a function of an externally applied DC current excitation emulating the net synaptic excitation current input to the neuron. In particular, the pair of complex-conjugate zeros move from a subcritical Hopf bifurcation point at low excitation current to a super-critical Hopf bifurcation point at high excitation current. The spikes are generated as the excitation current approaches the vicinity of the edge of chaos, which leads to the onset of the subcritical Hopf bifurcation regime. It follows from this in-depth qualitative analysis that local activity is the origin of spikes.

A theoretical study based on the application of Fokker–Planck theory to the dynamics of a cell membrane model is presented. The role and the meaning of nonlinear interaction of electric-field-sensitive channels is discussed in the framework of the coupled mode theory of weakly ionized media. A phenomenological model of Malthus-Verhulst type is derived for the size of activating channels of a single ion species.