Hodgkin Huxley model

Modified Hodgkin-Huxley Model. In the HH model, the membrane current I, written as a function of V is expressed by the system of coupled equations given in Table I. In these equations V is the displacement of membrane potential from the resting value (depolarization negative). Constants Cm, g, gjga, 8i VK VNa and V] are explained in detail in ( 1). In Table I, l n and g m h are the potassium and sodium conductances, respectively. The dimensionless dynamical quantities m, n, and h are solutions of the given first order differential equations and vary between zero and unity after a change in membrane potential. The a and 3 rate constants are assumed to depend only on the instantaneous value of membrane potential. 

Zhou, C., and Kurths, J. Noise-induced synchronization and coherence resonance of a Hodgkin-Huxley model of thermally sensitive neurons. Chaos 2003,13 401— 409. 

The 1952 Hodgkin-Huxley model for membrane electrical potential is perhaps the oldest and the best known cellular kinetic model that exhibits temporal oscillations. The phenomenon of the nerve action potential, also known as excitability, has grown into a large interdisciplinary area between biophysics and neurophysiology, with quite sophisticated mathematical modeling. See [103] for a recent treatise. 

The Hodgkin-Huxley model involves three membrane currents due to potassium, sodium, and a leak current of charge through other pathways. The model assumes linear current-voltage relationships ... 

The voltage v in the Hodgkin-Huxley model is the membrane potential measured relative to the equilibrium voltage Veq v = AT — Veq, where Veq is the potential when no current is applied. The experimentally determined equilibrium potentials (which depend on the ion gradients across the membrane) for the model are... 

Figure 7.7 Functions m< >, n00, and /too predicted by Equations (7.31) and (7.32) for the Hodgkin-Huxley model. Figure adapted from [108],...

A Matlab computer code for the Hodgkin-Huxley model is given below. The code to compute the time derivatives of the state variables (the right-hand side of Equation (7.30)) is ... 

Use computer simulation to determine if the solution to the Hodgkin-Huxley model of Section 7.3.3 is periodic at Iapp = 6.2 pA-cm 2. What happens when the applied current is lowered to 6.0 pA-citr2 ... 

As was recalled by Keynes (1994a), it was first observed by Chandler Meves (1970) that in squid axons perfused with NaF a small flow of Na ions persisted in the inactivated steady state. Their tentative conclusion based on the Hodgkin-Huxley model was to suppose that the inactivation parameter h was the sum of two components h, and Zij, where A, predominated at negative potentials, and Zij predominated at positive ones. This proposition predicted the existence of one type of Na+ conductance that increased transiently with depolarization as in the Hodgkin—Huxley system, and a second type that persisted with depolarization to give a steady low level of conductance. Combined with m kinetics the idea fitted well with the experimental data. 

The model of globally coupled oscillators is commonly used as a simplest model of neural synchrony. We illustrate this using a computationally efficient neuronal model, proposed by Rulkov [42, 43]. In this model a neuron is described by a 2D map. In spite of its simplicity, this model reproduces most regimes exhibited by the full Hodgkin-Huxley model, but at essentially lower computational costs, thus allowing detailed analysis of the dynamics of large ensembles. The model reads... 

The psychologist Maslow wrote that if the only tool you have is a hammer, you tend to treat everything as if it were a nail (4). Markov processes based on the Hodgkin-Huxley model had been widely used to describe ionic currents measured in many different experiments. However, in 1986, we began to use a new tool to analyze the patch clamp data. The insight gained from this new analysis has changed our ideas about the processes that open and close the ion channel. The new tool is based on fractals. 

The Markov description of ion channel kinetics, originally derived from the Hodgkin-Huxley model, implies that the ion channel protein has certain physical properties. 

Hassard, B. 1978. Bifurcation of periodic solutions of the Hodgkin-Huxley model for the squid giant axon. J. Theor. Biol. 71 401-20. 

P reflects the input resistance of the cell and was chosen such that the map model matches a corresponding equivalent Hodgkin-Huxley model (Traub and Miles 1991), see Figure 1.7. The details of the corresponding model can be found in (Nowotny et al. 2005). 

The 1000-fold speedup of the map model over a conventional Hodgkin-Huxley model allowed us to simulate learning in a system of approximately the size of the Drosophila olfactory pathway. The model system is illustrated in Figure 1.8. The... 

FIGURE 19.3 Nerve action potential. Computed with the Hodgkin-Huxley model for nerve membrane (membrane patch). B, baseline E, excitation E, plateau E, recovery A, afterpotential. 

Figure 4. Effect of potential on the parametric functions and in the Hodgkin-Huxley model (u = m,n, h).

Determined by the ratio of pyrene monomer to excimer fluorescence intensities. Hodgkin-Huxley model. 

A concrete example of a model for neuronal bursting (Plant, 1981) has been analyzed in considerable detail by Rinzel and Lee (1987). The model utihzes the Hodgkin-Huxley model, eqs. (13.3) and (13.4), as the mechanism for action-potential generation and introduces two additional conductances, a calcium channel and a calcium-activated potassium channel, to produce the bursting behavior. The membrane potential is given by... 

US to quite different excitation models, namely, a molecular model in the following section and a dissipative one in Section 6.2. We will come back to the Hodgkin-Huxley model in Section 5.6 in context with a generalized network language. 

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