Narrow Results

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.

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.

Homoclinic bifurcations of both equilibria and periodic orbits are argued to be critical for understanding the dynamics of the Hindmarsh–Rose model in particular, as well as of some square-wave bursting models of neurons of the Hodgkin–Huxley type. They explain very well various transitions between the tonic spiking and bursting oscillations in the model. We present the approach that allows for constructing Poincaré return mapping via the averaging technique. We show that a modified model can exhibit the blue sky bifurcation, as well as, a bistability of the coexisting tonic spiking and bursting activities. A new technique for localizing a slow motion manifold and periodic orbits on it is also presented.

We investigate the phase locking of firing events in periodically forced stochastic excitable systems. Our study is motivated by the observation of randomly phase locked firing activity in a large number of neurons, especially those involved in transducing physical stimuli such as temperature, sound, pressure and electric fields. The purpose of our paper is to review our work on the biophysical origin of such firing patterns. Special attention is given to the constructive effect of noise, and to the connection between stochastic and deterministic resonances in such systems. We present new results on stochastic phase locking in the subthreshold and suprathreshold regimes, and on the non-monotonicity of the spectral amplification as a function of noise intensity. We also discuss current outstanding problems and potentially fruitful future research directions.

Sufficient conditions for the global attractivity of the equilibrium point of the delay difference equations, which appears as models of the actions of neurons with dynamical threshold effects, are obtained by applying the method of Liapunov functionals and others.