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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 Θ⅜ (T° C ) cm/sec, [Formula: see text] 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.

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 max mV 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.

Current induced oscillations of a space clamped neuron action potential demonstrates a bifurcation scenario originally encapsulated by the four-dimensional Hodgkin–Huxley equations. These oscillations were subsequently described by the two-dimensional FitzHugh–Nagumo Equations in close agreement with the Hodgkin–Huxley theory. It is shown that the FitzHugh–Nagumo equations can to close approximation be reduced to a generalized van der Pol oscillator externally driven by the current. The current functions as an external constant force driving the action potential. As a consequence approximate analytic expressions are derived which predict the bifurcation scenario, the amplitudes of the oscillations and the oscillation periods in terms of the current and the physiological constants of the FitzHugh–Nagumo model. A second reduction permits explicit analytic solution and results in a spiking model which can be multiply coupled and extended to include the dynamics of phase locking, entrainment and chaos characteristic of time-dependent synaptic inputs.

Relaxation oscillations appear in processes which involve transitions between two states characterized by fast and slow time scales. When a relaxation oscillator is coupled to an external periodic force its entrainment by the force results in a response which can include multiple periodicities and bistability. The prototype of these behaviors is the harmonically driven van der Pol equation which displays regions in the parameter space of the driving force amplitude where stable orbits of periods 2n ± 1 coexist, flanked by regions of periods 2n + 1 and 2n - 1. The parameter regions of such bistable orbits are derived analytically for the closely related harmonically driven Stoker–Haag piecewise discontinuous equation. The results are valid over most of the control parameter space of the system. Also considered are the reasons for the more complicated dynamics featuring regions of high multiple periodicity which appear like noise between ordered periodic regions. Since this system mimics in detail the less analytically tractable forced van der Pol equation, the results suggest extensions to situations where forced relaxation oscillations are a component of the operating mechanisms.

Relaxation oscillations are characteristic of periodic processes consisting of segments which differ greatly in time: a long-time span when the system is moving slowly and a relatively short time span when the system is moving rapidly. The period of oscillation, the sum of these contributions, is usually treated by singular perturbation theory which starts from the premise that the long span is asymptotically extended and the short span shrinks asymptotically to a single instant. Application of the theory involves the analysis of adjacent dynamical regions and multiple time scales. The relaxation oscillations of the Stoker–Haag piecewise-linear discontinuous equation and the van der Pol equation are investigated using a simpler analytical method requiring only the connection at a point of the two dynamical fast and slow regions. Compared to the results of singular perturbation theory, the quantitative results of the present method are more accurate in the Stoker–Haag case and marginally less accurate in the van der Pol case. The relative simplicity of the formulation suggests extension to three-dimensional systems where relaxation oscillations can become unstable leading to bistability, multiple periodicity and chaos.

Oscillations described by autonomous three-dimensional differential equations display multiple periodicities and chaos at critical parameter values. Regardless of the subsequent scenario the key instability is often an initial bifurcation from a single period oscillation to either its subharmonic of period two, or a symmetry breaking bifurcation. A generalized third-order nonlinear differential equation is developed which embraces the dynamics vicinal to these bifurcation events. Subsequently, an asymptotic averaging formalism is applied which permits an analytic development of the bifurcation dynamics, and, within quantifiable limits, prediction of the instability of the period one orbit in terms of the system control parameters. Illustrative applications of the general formalism, are made to the Rössler equations, Lorenz equations, three-dimensional replicator equations and Chua's circuit equations. The results provide the basis for discussion of the class of systems which fall within the framework of the formalism.

Oscillations described by autonomous three-dimensional differential equation systems display multiple periodicities and chaos at critical parameter values. Regardless of the subsequent scenario, the key instability is usually an initial bifurcation from a single period oscillation to its subharmonic of period two, or the reverse. An asymptotic averaging formalism is introduced by means of an example which permits an analytic development of the bifurcation dynamics, and in particular, prediction of the onset of periods 1 ↔ 2 bifurcations in terms of the system control parameters.

One-dimensional return maps characterized by one critical point produce a universal sequence of periodic orbits, the Metropolis, Stein & Stein (MSS) sequence. Iterates of such maps can be regarded as an object whose structure is defined by the MSS sequence. The two critical point mapping [Formula: see text] as the control parameter b varies between [Formula: see text] and 1 is shown to display evolution from a single MSS structure to two identical MSS structures, suggestive of reproduction. The process of reproduction connotes a causal relationship whereby an original structure provides the blueprint for a copy. Here causality is replaced by an omniscient instruction, the mapping, and reproduction is achieved by progressively changing the instruction through variation of the control parameter. Bistability plays a crucial role in map dynamics of reproduction, which is viewed as a holistic process providing an alternative mechanism to digital replication.