Modeling of time disparity detection by the Hodgkin-Huxley equations

Phase-sensitive neurons in the electrosensory lateral line lobe in the electrosensory pathway of the wave-type electric fish, Gymnarchus niloticus, are specialized for sensing the time disparity between sensory inputs at different parts of the body surface that is necessary for an electrical behavior, jamming avoidance response. These neurons are sensitive to time disparity in the microsecond range between synaptic inputs that represent occurrence times of electrosensory signals at different areas on the body surface. We showed that an ideal Hodgkin-Huxley equation may serve as a time disparity detector that fits physiological precision, and the precision for the time disparity detection is largely regulated by the maximal g(K) conductance in the Hodgkin-Huxley equations.     

Isolated periodic solutions of the Hodgkin-Huxley equations

Periodic solutions of the current clamped Hodgkin-Huxley equations (Hodgkin & Huxley, 1952 J. Physiol. 117, 500) that arise by degenerate Hopf bifurcation were studied recently by Labouriau (1985 SIAM J. Math. Anal. 16, 1121, 1987 Degenerate Hopf Bifurcation and Nerve Impulse (Part II), in press). Two parameters, temperature T and sodium conductance gNa were varied from the original values obtained by Hodgkin & Huxley. Labouriau's work proved the existence of small amplitude periodic solution branches that do not connect locally to the stationary solution branch, and had not been previously computed. In this paper we compute these solution branches globally. We find families of isolas of periodic solutions (i.e. branches not connected to the stationary branch). For values of gNa in the range measured by Hodgkin & Huxley, and for physically reasonable temperatures, there are isolas containing orbitally asymptotically stable solutions. The presence of isolas of periodic solutions suggests that in certain current space clamped membrane experiments, action potentials could be observed even though the stationary state is stable for all current stimuli. Once produced, such action potentials will disappear suddenly if the current stimulus is either increased or decreased past certain values. Under some conditions, "jumping" between action potentials of different amplitudes might be observed.     

Loss of synchronization in partially coupled Hodgkin-Huxley equations

We study the loss of synchronization of two partially coupled space-clamped Hodgkin-Huxley equations, with symmetric coupling. This models the coupling of two cells through an electrical synapse. For strong enough coupling it is known that all solutions of the equations approach a state where the two cells are perfectly synchronized, having the same behaviour at each moment. We describe the local bifurcations that arise when the coupling strength is reduced, using a mixture of analytical and numerical methods. We find that perfect synchrony is retained for very small positive values of the coupling strength, for almost all initial conditions. Although perfect synchrony is lost for negative values of the coupling constant, the system always retains some degree of synchronization until it becomes totally unstable. This happens in two ways: in many cases for almost all initial conditions the solutions still approach a perfectly synchronized state. Even when this is not true, the attracting solutions are still synchronized, with a half-period phase shift.     

Canards for a reduction of the Hodgkin-Huxley equations

This paper shows that canards, which are periodic orbits for which the trajectory follows both the attracting and repelling part of a slow manifold, can exist for a two-dimensional reduction of the Hodgkin-Huxley equations. Such canards are associated with a dramatic change in the properties of the periodic orbit within a very narrow interval of a control parameter. By smoothly connecting stable and unstable manifolds in an asymptotic limit, we predict with great accuracy the parameter value at which the canards exist for this system. This illustrates the power of using singular perturbation theory to understand the dynamical properties of realistic biological systems.     

Coincidence detection in phosphoinositide signaling

Phosphoinositide lipids function as both signaling molecules and as compartment-specific localization signals for phosphoinositide-binding proteins. In recent years, both phosphoinositides and phosphoinositide-binding proteins have been reported to display a restricted, rather than a uniform, distribution across intracellular membranes. Here, we examine recent data documenting the restricted distribution of both phosphoinositides and phosphoinositide-binding proteins and examine how phosphoinositide-binding proteins might engage multiple binding partners to achieve these restricted localizations, effectively acting as detectors of coincident localization signals.     

Complex nonlinear dynamics of the Hodgkin-Huxley equations induced by time scale changes

 The Hodgkin–Huxley equations with a slight modification are investigated, in which the inactivation process (h) of sodium channels or the activation process of potassium channels (n) is slowed down. We show that the equations produce a variety of action potential waveforms ranging from a plateau potential, such as in heart muscle cells, to chaotic bursting firings. When h is slowed down – differently from the case of n variable being slow – chaotic bursting oscillations are observed for a wide range of parameter values although both variables cause a decrease in the membrane potential. The underlying nonlinear dynamics of various action potentials are analyzed using bifurcation theory and a so-called slow–fast decomposition analysis. It is shown that a simple topological property of the equilibrium curves of slow and fast subsystems is essential to the production of chaotic oscillations, and this is the cause of the large difference in global firing characteristics between the h-slow and n-slow cases. Received: 9 August 2000 / Accepted in revised form: 10 January 2001     

Homomorphism on a physical system of the Hodgkin-Huxley equations for ion conductance in nerve

A homomorphism on a physical system of the Hodgkin-Huxley equations for ion conductance in nerve is derived. It is pointed out that a homomorphism can correct the Cole-Moore discrepancy in delay of conductance for voltage clamp data with initial hyperpolarization. The voltage dependence of the rate constants can also be removed. Curves are presented to compare the representation of the nerve conductances by the Hodgkin-Huxley equations and the new homomorphism.     

A note on the asymptotic reduction of the Hodgkin-Huxley equations for nerve impulses

The (standard) FitzHugh reduction of the Hodgkin-Huxley equations for the propagation of nerve impulses ignores the dynamics of the activation gates. This assumption is invalid and leads to an over-estimation of the wave speed by a factor of 5 and the wrong dependence of wave speed on sodium channel conductance. The error occurs because a non-dimensional parameter, which is assumed to be small in the FitzHugh reduction, is in fact large (≈18). We analyse the Hodgkin-Huxley equations for propagating nerve impulses in the limit that this non-dimensional parameter is large, and show that the analytical results are consistent with numerical simulations of the Hodgkin-Huxley equations.     

Intermittency in the Hodgkin-Huxley model

We show that action potentials in the Hodgkin-Huxley neuron model result from a type I intermittency phenomenon that occurs in the proximity of a saddle-node bifurcation of limit cycles. For the Hodgkin-Huxley spatially extended model, describing propagation of action potential along axons, we show the existence of type I intermittency and a new type of chaotic intermittency, as well as space propagating regular and chaotic diffusion waves. Chaotic intermittency occurs in the transition from a turbulent regime to the resting regime of the transmembrane potential and is characterised by the existence of a sequence of action potential spikes occurring at irregular time intervals.     

Application of the Hodgkin-Huxley equations to an electric field model for interaction between excitable cells

We previously described a model for the electrical transfer of excitation from one cell to the next which utilized the electric potential generated in the junctional cleft between the cells. Low-resistance connections between the cells were not used in the model, and it was assumed that the junctional membranes were excitable. This model was analyzed for the static case without capacitances and for the dynamic case in which capacitances were part of the circuit elements. For simplicity, the Na⁺ resistance (R_Na), after a threshold potential was exceeded, was allowed to decrease exponentially (to 1% of its initial value) within 0·25–1·0 ms, and possible changes in the K⁺ resistance were ignored. In this paper, we have incorporated the Hodgkin-Huxley equations into the operation of the lumped membrane units for the electrical equivalent circuit of the cell membrane. The parameters varied are the membrane capacitances, resistances, maximum Na⁺ conductance ($\text{g}{}_{\mathrm{<mtext>Na</mtext>}}$), and the radial cleft resistance (R_jc). We demonstrated that our model worked very well, i.e. the successful transfer of action potentials was achieved, with the membrane units following Hodgkin-Huxley dynamics for changes in g_Na and g_K. The calculations indicate that transmission is facilitated when the junctional units have a higher $\text{g}{}_{\mathrm{<mtext>Na</mtext>}}$ and a lower capacitance and when R_jc is elevated. Lowering the resistance of the junctional membrane units several fold, relative to the surface membrane units, also facilitated transmission; however, the absolute resistance of the junctional membrane was still well above the maximum value that would allow sufficient local-circuit current to flow to effect transmission. Thus, the electric field model provides an alternative means of cell-to-cell propagation between myocardial cells which is electrical in nature but does not require the presence of low-resistance connections between cells.     

Spikes annihilation in the Hodgkin-Huxley neuron

The Hodgkin–Huxley (HH) neuron is a nonlinear system with two stable states: A fixed point and a limit cycle. Both of them co-exist. The behavior of this neuron can be switched between these two equilibria, namely spiking and resting respectively, by using a perturbation method. The change from spiking to resting is named Spike Annihilation, and the transition from resting to spiking is named Spike Generation. Our intention is to determine if the HH neuron in 2D is controllable (i.e., if it can be driven from a quiescent state to a spiking state and vice versa). It turns out that the general system is unsolvable.¹ In this paper, first of all,² we analytically prove the existence of a brief current pulse, which, when delivered to the HH neuron during its repetitively firing state, annihilates its spikes. We also formally derive the characteristics of this brief current pulse. We then proceed to explore experimentally, by using numerical simulations, the properties of this pulse, namely the range of time when it can be inserted (the minimum phase and the maximum phase), its magnitude, and its duration. In addition, we study the solution of annihilating the spikes by using two successive stimuli, when the first is, of its own, unable to annihilate the neuron. Finally, we investigate the inverse problem of annihilation, namely the spike generation problem, when the neuron switches from resting to firing. ¹ This conclusion is a consequence of three well-known fundamental results, namely Hilbert 16th Problem, the Poincare–Bendixon Theorem and the Hopf Bifurcation Theorem. ² We are extremely grateful to the feedback we received from the anonymous Referees to the initial version of the paper. Their comments significantly improved the quality of the current version. Thanks a lot!     

Hopf bifurcations in multiple-parameter space of the Hodgkin-Huxley equations I. Global organization of bistable periodic solutions

 The Hodgkin-Huxley equations (HH) are parameterized by a number of parameters and shows a variety of qualitatively different behaviors depending on the parameter values. We explored the dynamics of the HH for a wide range of parameter values in the multiple-parameter space, that is, we examined the global structure of bifurcations of the HH. Results are summarized in various two-parameter bifurcation diagrams with I _ext (externally applied DC current) as the abscissa and one of the other parameters as the ordinate. In each diagram, the parameter plane was divided into several regions according to the qualitative behavior of the equations. In particular, we focused on periodic solutions emerging via Hopf bifurcations and identified parameter regions in which either two stable periodic solutions with different amplitudes and periods and a stable equilibrium point or two stable periodic solutions coexist. Global analysis of the bifurcation structure suggested that generation of these regions is associated with degenerate Hopf bifurcations. Received: 23 April 1999 / Accepted in revised form: 24 September 1999     

Mathematical description of a bursting pacemaker neuron by a modification of the Hodgkin-Huxley equations.

Modifications based on experimental results reported in the literature are made to the Hodgkin-Huxley equations to describe the electrophysiological behavior of the Aplysia abdominal ganglion R15 cell. The system is then further modified to describe the effects with the application of the drug tetrodotoxin (TTX) to the cells' bathing medium. Methods of the qualitative theory of differential equations are used to determine the conditions necessary for such a system of equations to have an oscillatory solution. A model satisfying these conditions is shown to preduct many experimental observations of R15 cell behavior. Numerical solutions are obtained for differential equations satisfying the conditions of the model. These solutions are shown to have a form similar to that of the bursting which is characteristic of this cell, and to preduct many results of experiments conducted on this cell. The physiological implications of the model are discussed.     

The singularly perturbed Hodgkin-Huxley equations as a tool for the analysis of repetitive nerve activity

A qualitative analysis of the Hodgkin-Huxley model (Hodgkin and Huxley 1952), which closely mimics the ionic processes at a real nerve membrane, is performed by means of a singular perturbation theory. This was achieved by introducing a perturbation parameter that, if decreased, speeds up the fast variables of the Hodgkin-Huxley equations (membrane potential and sodium activation), whereas it does not affect the slow variables (sodium inactivation and potassium activation). In the most extreme case, if the perturbation parameter is set to zero, the original four-dimensional system degenerates to a system with only two differential equations. This degenerate system is easier to analyze and much more intuitive than the original Hodgkin-Huxley equations. It shows, like the original model, an infinite train of action potentials if stimulated by an input current in a suitable range. Additionally, explanations for the increased sensitivity to depolarizing current steps that precedes an action potential can be found by analysis of the degenerate system. Using the theory of Mishchenko and Rozov (1980) it is shown that the degenerate system does not only represent a simplification of the original Hodgkin-Huxley equations but also gives a valid approximation of the original model at least for stimulating currents that are constant within a suitable range.     

Action Potential Initiation in the Hodgkin-Huxley Model

A recent paper of B. Naundorf et al. described an intriguing negative correlation between variability of the onset potential at which an action potential occurs (the onset span) and the rapidity of action potential initiation (the onset rapidity). This correlation was demonstrated in numerical simulations of the Hodgkin-Huxley model. Due to this antagonism, it is argued that Hodgkin-Huxley-type models are unable to explain action potential initiation observed in cortical neurons in vivo or in vitro. Here we apply a method from theoretical physics to derive an analytical characterization of this problem. We analytically compute the probability distribution of onset potentials and analytically derive the inverse relationship between onset span and onset rapidity. We find that the relationship between onset span and onset rapidity depends on the level of synaptic background activity. Hence we are able to elucidate the regions of parameter space for which the Hodgkin-Huxley model is able to accurately describe the behavior of this system.     

Hopf bifurcations in multiple-parameter space of the Hodgkin-Huxley equations II. Singularity theoretic approach and highly degenerate bifurcations

In the Hodgkin-Huxley equations (HH), we have identified the parameter regions in which either two stable periodic solutions with different amplitudes and periods and an equilibrium point or two stable periodic solutions coexist. The global structure of bifurcations in the multiple-parameter space in the HH suggested that the bistabilities of the periodic solutions are associated with the degenerate Hopf bifurcation points by which several qualitatively different behaviors are organized. In this paper, we clarify this by analyzing the details of the degenerate Hopf bifurcations using the singularity theory approach which deals with local bifurcations near a highly degenerate fixed point. Received: 23 April 1999 / Accepted in revised form: 24 September 1999