Voltage dependence of Hodgkin-Huxley rate functions for a multistage<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>K</mml:mi><mml:mo>+</mml:mo></mml:msup></mml:math>channel voltage sensor within a membrane

Vaccaro, S. R.

doi:10.1103/physreve.90.052713

Cited by 3 publications

(12 citation statements)

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“…and α n and β n are voltage dependent rate functions. This equation may be derived from a kinetic model for K+ channel gating where the voltage dependence of α n and β n may be expressed in terms of the transition rates for a two stage voltage sensor activation process [22,23]. Therefore, the membrane current equation is…”

Section: Reduction Of a Kinetic Model For Na+ Chan-nel Activation Andmentioning

confidence: 99%

Reduction of a kinetic model for Na+ channel activation, and fast and slow inactivation within a neural or cardiac membrane

Vaccaro

2019

Phys. Rev. E

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A fifteen state kinetic model for Na+ channel gating that describes the coupling between three activation sensors, a twostage fast inactivation process and slow inactivated states, may be reduced to equations for a six state system by application of the method of multiple scales. By expressing the occupation probabilities for closed states and the open state in terms of activation and fast inactivation variables, and assuming that activation has a faster relaxation than inactivation and that the activation sensors are mutually independent, the kinetic equations may be further reduced to rate equations for activation, and coupled fast and slow inactivation that describe spike frequency adaptation, a repetitive bursting oscillation in the neural membrane, and a cardiac action potential with a plateau oscillation. The fast inactivation rate function is, in general, dependent on the activation variable m(t) but may be approximated by a voltage-dependent function, and the rate function for entry into the slow inactivated state is dependent on the fast inactivation variable. 1 INTRODUCTIONDuring prolonged or repetitive depolarization, in addition to the fast inactivation of Na channels that contributes to repolarization of the membrane [1], a slow inactivation process reduces the number of Na+ channels available for activation. The increase in slow inactivation of Na+ channels during depolarization is associated with a delay to the next spike or a reduction in the firing frequency (spike frequency adaptation) [2] and is the result of a structural rearrangement in the selectivity filter region of the ion channel that generally occurs following the inactivation of the pore [3]. Slow inactivation of the transient and persistent components of the Na+ current in a mesencephalic V neuron is associated with the termination of a bursting oscillation, and the increase in the amplitude of the subthreshold oscillation between bursts occurs during the recovery from slow inactivation [4]. In subicular neurons adjacent to the hippocampus, the transition from bursting to single spiking is influenced by the slow inactivation of Na+ channels, and this may provide a mechanism for enhancing the effect of input signals [5].For Na+ channels with slow inactivation, the Na+ current I N a may be described by the expression m 3 hs(V N a − V ) [2] where V N a is the equilibrium potential, and the activation variable m, the fast inactivation variable h, and the slow inactivation variable s satisfy the equations

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Section: Reduction Of a Kinetic Model For Na+ Chan-nel Activation Andmentioning

confidence: 99%

Reduction of a kinetic model for Na+ channel activation, and fast and slow inactivation within a neural or cardiac membrane

Vaccaro

2019

Phys. Rev. E

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“…If the deinactivation rates σ 1 > σ 2 > σ 3 ≈ 0 are chosen to satisfy microscopic reversibility, it may be shown from the numerical solution of the master equation or from a more general form of the solution of Eqs. (18) to (23) that O(t) = m(t) 2 h(t) is still a good approximation. From Eq.…”

Section: Voltage Clamp Of a Na+ Channel With Two Ac-tivation Sensorsmentioning

confidence: 99%

“…It is assumed that the transition rates for each stage of inactivation are dependent on single barrier activation, and therefore, are proportional to exp(-U) where U is the voltage dependent height of the barrier [17]. However, if the Na+ channel S4 sensors of the DI, DII or DIII domains are activated in two stages, the rate functions α m and β m may be approximated by two-state expressions [18].…”

Section: Voltage Clamp Of a Na+ Channel With Two Ac-tivation Sensorsmentioning

confidence: 99%

“…, and the first forward and backward transitions are rate limiting ( β ik ≫ δ ik and γ ik ≫ α ik , for k = 1 to 3) [18,19], the occupation probabilities A 1 , A 2 and A 3 rapidly attain a quasi steady state…”

Section: Voltage Clamp Of a Na+ Channel With Two Ac-tivation Sensorsmentioning

confidence: 99%

“…4) [18]. If the DIII S4 sensor is the slowest to deactivate (β B1 ≪ β B2 ) [9,10], ω 1G and ω 2G are solutions of the equation…”

Section: Voltage Clamp Of a Na+ Channel With Two Ac-tivation Sensorsmentioning

confidence: 99%

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Derivation of Hodgkin-Huxley equations for aNa+channel from a master equation for coupled activation and inactivation

Vaccaro¹

2016

Phys. Rev. E

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The Na+ current in nerve and muscle membranes may be described in terms of the activation variable m(t) and the inactivation variable h(t), which are dependent on the transitions of S4 sensors of each of the Na+ channel domains DI to DIV. The time-dependence of the Na+ current and the rate equations satisfied by m(t) and h(t) may be derived from the solution to a master equation which describes the coupling between two or three activation sensors regulating the Na+ channel conductance and a two stage inactivation process. If the inactivation rate from the closed or open states increases as the S4 sensors activate, a more general form for the Hodgkin-Huxley expression for the open state probability may be derived where m(t) is dependent on both activation and inactivation processes. The voltage dependence of the rate functions for inactivation and recovery from inactivation are consistent with the empirically determined expressions, and exhibit saturation for both depolarized and hyperpolarized clamp potentials. 1 INTRODUCTIONThe opening and subsequent inactivation of Na+ channels and the activation of K+ channels generate the action potential in nerve and muscle membranes [1]. The time-dependence of the Na+ current in the squid axon may be described in terms of the expression m(t) 3 h(t) where the activation variable m(t) and inactivation variable h(t) satisfy the rate equations

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Rate equations for a Na+ channel gating master equation during the action potential within a neural membrane

Vaccaro¹

2017

Preprint

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The action potential in a neural membrane is generated by Na+ and K+ channel ionic currents that may be calculated from a current equation and the rate equations for activation variables m and n, and the Na+ inactivation variable h. Assuming that a Na+ channel has three activation sensors, and activation and inactivation are cooperative processes, a twelve state master equation that describes channel gating may be reduced to kinetic equations for a five state system when the occupational probability of the first inactivated state is small, and the remaining inactivated states contribute to a total inactivated state. In the case of independent activation sensors, the inactivation rate is, in general, dependent on the activation variable m(t) as well as the forward inactivation transition rates. However, when m(t) has a faster time constant than h(t), the inactivation rate may be approximated by a voltage-dependent function, and therefore, the solution of the master equation during an action potential may be approximated by the solution of Hodgkin-Huxley rate equations for m and h.

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Voltage dependence of Hodgkin-Huxley rate functions for a multistageK+channel voltage sensor within a membrane

Cited by 3 publications

References 51 publications

Reduction of a kinetic model for Na+ channel activation, and fast and slow inactivation within a neural or cardiac membrane

Reduction of a kinetic model for Na+ channel activation, and fast and slow inactivation within a neural or cardiac membrane

Derivation of Hodgkin-Huxley equations for aNa+channel from a master equation for coupled activation and inactivation

Rate equations for a Na+ channel gating master equation during the action potential within a neural membrane

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