Hodgkin-Huxley System Research Articles

Finite Dimensionality and Upper Semicontinuity of the Global Attractor of Singularly Perturbed Hodgkin–Huxley Systems

In an effort to show that the standard Hodgkin–Huxley system approximates the hyperbolic Hodgkin–Huxley system we view the hyperbolic system as a singular perturbation of the standard system. We establish the existence of global attractors for the singularly perturbed hyperbolic system and show that they have finite fractal and Hausdorff dimensions. We then show that the global attractors of the singularly perturbed system converge to the global attractor for the standard system as the small perturbation parameter decreases to zero.

A special point of Z2-codimension three Hopf bifurcation in the Hodgkin-Huxley model

A curve in ( I, T, g Na , v Na )-parameter space of degenerate Hopf bifurcation for the Hodgkin-Huxley model is described. The curve includes all six of the “doubly degenerate” points from our earlier work [1,2]. The curve includes one special point at which m = 1, where m is the modal parameter. The recognition problem for the singularity at this point is solved. Assuming that the parameters ( T, g Na , v Na ) provide a versal unfolding, all previously known types of Hopf bifurcation behavior in the Hodgkin-Huxley system will be exhibited for ( T, g Na , v Na ) in a neighborhood of their values at this point.

A PROBABILISTIC MODEL OF NEURONAL INTEGRATION

A probabilistic estimation of an integrate and fire model commonly used for performing biologically plausible neurocomputing is proposed. It processes the temporal information of the input stimuli as a group of cells — the stellate or Chopper cells of the cochlear nucleus. The main properties are the same as the ones of the common Hodgkin–Huxley system. This model has a limited set of parameters: time constant of the membrane potential, threshold level, refractory durations. It is defined by a delay differential system having a low cost of computation. Original properties such as blocking and reset are introduced, providing a control of the oscillatory behaviour useful for building functional neural nets. In order to recover the parameters of neurons, we also analyze the first interspike interval histogram.

Changes in sensitivities of amplitude and frequency curves of the Hodgkin-Huxley system produced by variations of maximal conductances

The question of sensitivity of amplitude and frequency curves of the Hodgkin-Huxley system to variations of maximal values of sodium ( g Na ) and potassium ( g K ) conductances is considered. Changes in the sensitivity produced by simultaneous changes in g Na and g K values are examined. It has been shown that simultaneous increase of the values of g Na and g k is accompanied by decrease of the sensitivity and, therefore, by improvement of quality of nerve fiber functioning.

Partition of the Hodgkin-Huxley type model parameter space into the regions of qualitatively different solutions.

We have examined the problem of obtaining relationships between the type of stable solutions of the Hodgkin-Huxley type system, the values of its parameters and a constant applied current (I). As variable parameters of the system the maximal Na+(-gNa), K+(-gK) conductances and shifts (Gm, Gh, Gn) of the voltage-dependences have been chosen. To solve this problem it is sufficient to find points belonging to the boundary, partitioning the parameter space of the system into the regions of the qualitatively different types of stable solutions (steady states and stable periodic oscillations). Almost all over the physiological range of I, a type of stable solution is determined by the type of steady state (stable or unstable). Using this fact, the approximate solution of this problem could be obtained by analyzing the spectrum of eigenvalues of the Jacobian matrix for the linearized system. The families of the plan sections of the boundary have been constructed in the three-parameter spaces (I, -gNa, -gK), (I, Gm, Gh), (I, Gm, Gn).

Reduction of conductance-based neuron models.

We present a scheme for systematically reducing the number of differential equations required for biophysically realistic neuron models. The techniques are general, are designed to be applicable to a large set of such models and retain in the reduced system as high a degree of fidelity to the original system as possible. As examples, we provide reductions of the Hodgkin-Huxley system and the A-current model of Connor et al. (1977).

The integro-differential equation form of a Hodgkin-Huxley type system of differential equations

A theorem is presented that establishes an equivalence between a Hodgkin-Huxley system of four first-order nonlinear ordinary differential equations (with initial values) and one first-order nonlinear ordinary integrodifferential equation (with initial value). The system was studied theoretically and numerically in [1].

Evidence for two types of sodium conductance in axons perfused with sodium fluoride solution.

1. Voltage clamp experiments were carried out on squid giant axons internally perfused with 300 mM-NaF + sucrose. K-free artificial sea-water, -0.3 to 3.5 degrees C, was used externally.2. Membrane currents were corrected for capacitative and leakage components, and the resulting Na current was converted to Na conductance, g(Na). An attempt was made to fit changes in g(Na) according to the Hodgkin-Huxley model, namely [Formula: see text]. According to the model g(Na) is a constant, m(infinity) and h(infinity) are steady-state values which depend only on voltage, tau(m) (-1) and tau(h) (-1) are rate constants which also are functions only of voltage.3. Stepwise depolarizations from the holding potential (-67 to -83 mV) to a potential which varied from -10 to +63 mV resulted in an exponential decline of h from its initial level to a final, non-zero level. If the test depolarization was preceded by a positive prepulse (duration, 19-105 msec; voltage, -6 to 94 mV) the rate constant for h, tau(h) (-1), was increased roughly threefold with practically no change in the final level.4. The steady-state level of h was studied by using prepulses of varying amplitude followed by a test depolarization. In one such experiment a value of 0.34 was obtained for a 105 msec prepulse to -49 mV. The same value for the steady level of h was obtained from analysing a record taken at +52 mV. If the potential was switched from -49 to +52 mV there was a transient increase in g(Na) although h(infinity) had the same value at these two potentials.5. Recovery from depolarization was studied by repolarizing the fibre for varying lengths of time, then applying a test depolarization. If the first depolarization was strongly positive (for example, 70 mV), so that the steady level of h was large (0.39), the currents associated with the test pulse could not be fitted on the basis of an exponential increase in h during the recovery period. Rather, the results suggested that on repolarization h rapidly decreased initially, then slowly increased.6. These results can be explained by assuming that h is given by the sum of two components, h(1) and h(2). Changes are represented kinetically by h(1) right harpoon over left harpoon x right harpoon over left harpoon h(2), where x signifies the inactive state. The distribution is shifted to the left at negative potentials and to the right for positive ones. The resulting Na conductance is comprised of two types: the first type, g(Na)m(3)h(1), is similar to the Hodgkin-Huxley system and underlines the usual transient increase in g(Na) associated with depolarization; the second type, g(Na)m(3)h(2), is maintained with depolarization and gives rise to a steady level of g(Na).

Boundedness theorems and other mathematical studies of a Hodgkin-Huxley-type system of differential equations: Numerical treatment of thresholds and stationary values: Parts I, II and III

We present here a study of the Hodgkin-Huxley system of four first-order nonlinear ordinary differential equations (with initial values) as reformulated by H.M. Lieberstein. The original Hodgkin-Huxley system modeled the development and propagation of voltage impulses on nerve fibers. It was altered by Dr. Lieberstein so as to include the effects of line inductance and line capacitance without introducing any new parameters. It included the application of a sustained constant membrane current density I 0. The inclusion of I 0 enables the system to model the electrical behavior of smooth muscle cells, pacemaker cells, receptor cells, and cells with a plateau-type behavior, in addition to nerve cells. In Part I, after a section that provides background material and preliminary facts concerning the system, local existence, uniqueness, and analyticity of solutions to the system are proved. Boundedness of the solutions is proved in Part II. It isproved there that three of the four solution components ( n, m, and h) have lower bounds of zero and upper bounds of one. The lower boundedness of the fourth component, the membrane voltage v, is then proved, followed by a proof for upper boundedness. A second proof for lower boundedness is also given. Boundedness of the solutions leads to the existence and uniqueness of solutions for all nonnegative time. The continuous dependence of the solutions on initial values and parameters follows. Besides proving that the sensitivity to parameters inherent in the original Hodgkin-Huxley model is removed, continuous dependence also limits the mathematical possibilities that could be involved in the threshold phenomenon that appears in the equations. New bounds for n, m, and h are also obtained from the fact that v is bounded. In Part III, the mathematical possibilities that might govern the threshold phenomena that appear in the model are examined and new numerical work is presented, which indicates that the model yields solutions that are intermediate to the “on” and “off” solutions expected with such phenomena. Then theorems and numerical work are presented indicating that stationary points are not involved in the threshold phenomenon, although they do determine the behavior of the system at large positive times.