A simple modification of the Hodgkin and Huxley equations explains type 3 excitability in squid giant axons

Clay, John R.; Paydarfar, David; Forger, Daniel B.

doi:10.1098/rsif.2008.0166

Cited by 71 publications

(59 citation statements)

References 39 publications

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“…In some experiments the squid giant axons had type 3 instead of type 2 excitability [16,17]. The discrepancy between the type 2 behavior of the HH model and experiment was explained by modifying a single parameter in the term describing the potassium current [18]. In a recent study of the periodically stimulated HH model by the present author it was found that the ring rate may be either continuous or discontinuous function of the current amplitude, depending on the stimulus frequency [19].…”

Section: Introductionmentioning

confidence: 99%

Multimodal Transition and Excitability of a Neural Oscillator

Borkowski

2012

Acta Phys. Pol. A

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We analyze the response of the MorrisLecar model to a periodic train of short current pulses in the period-amplitude plane. For a wide parameter range encompassing both class 2 and class 3 behavior in the Hodgkin classication there is a multimodal transition between the set of odd modes and the set of all modes. It is located between the 2:1 and 3:1 locked-in regions. It is the same dynamic instability as the one discovered earlier in the HodgkinHuxley model and observed experimentally in squid giant axons. It appears simultaneously with the bistability of the states 2:1 and 3:1 in the perithreshold regime. These results imply that the multimodal transition may be a universal property of resonant neurons.

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Section: Introductionmentioning

confidence: 99%

Multimodal Transition and Excitability of a Neural Oscillator

Borkowski

2012

Acta Phys. Pol. A

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“…In a recent study, Paydarfar, Forger and Clay (2006) showed that small oscillatory stimuli can be used to induce a state transition in a bistable system. While, we could theoretically model the Hodgkin-Huxley model as a bistable system by adding a sufficiently large exogenous depolarizing persistent current, previous studies have shown that the squid axon, which is the basis for the Hodgkin Huxley model, fails to exhibit repetitive firing under the condition of a persistent depolarizing current clamp (Clay, Paydarfar, & Forger 2008). Furthermore, our preliminary analysis suggested that the first-order gradient algorithm does not readily converge because of the sensitivity of the Hodgkin-Huxley system, specifically the terminal point, even to small changes in the stimulus.…”

Section: Methodsmentioning

confidence: 99%

Switching neuronal state: optimal stimuli revealed using a stochastically-seeded gradient algorithm

Chang

Paydarfar

2014

J Comput Neurosci

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Inducing a switch in neuronal state using energy optimal stimuli is relevant to a variety of problems in neuroscience. Analytical techniques from optimal control theory can identify such stimuli; however, solutions to the optimization problem using indirect variational approaches can be elusive in models that describe neuronal behavior. Here we develop and apply a direct gradient-based optimization algorithm to find stimulus waveforms that elicit a change in neuronal state while minimizing energy usage. We analyze standard models of neuronal behavior, the Hodgkin-Huxley and FitzHugh-Nagumo models, to show that the gradient-based algorithm: 1) enables automated exploration of a wide solution space, using stochastically generated initial waveforms that converge to multiple locally optimal solutions; and 2) finds optimal stimulus waveforms that achieve a physiological outcome condition, without a priori knowledge of the optimal terminal condition of all state variables. Analysis of biological systems using stochastically-seeded gradient methods can reveal salient dynamical mechanisms underlying the optimal control of system behavior. The gradient algorithm may also have practical applications in future work, for example, finding energy optimal waveforms for therapeutic neural stimulation that minimizes power usage and diminishes off-target effects and damage to neighboring tissue.

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“…Na + currents were as described above using the PC kinetic scheme (Eq. (4)); K + and leakage currents were as per HH (respective maximal conductances: 36 and 0.3 mS/cm 2 ; respective equilibrium potentials: −72 and −49.4 mV, [36]; cf., [12,48]). Ionic concentrations were assumed to remain constant (cf., [60,65]); channel densities were assumed to be uniformly distributed with no clustering over the axonal surface area (cf.…”

Section: Numerical Analysis and Axonal Cablementioning

confidence: 96%

Probability Distributions of Markovian Sodium Channel States During Propagating Axonal Impulses With or Without Recovery Supernormality

Goldfinger

2009

J. Integr. Neurosci.

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This study addressed a macroscopic neurophysiological phenomenon - supernormality during the recovery phase of propagating axonal impulses - in explicit chemical terms. Excitation was reconstructed numerically using the kinetic scheme of multiple-state probabilistic transitions within a population of voltage-dependent sodium channels (NaCh) derived by Vandenberg and Bezanilla ("PC" scheme). Each NaCh transition was characterized as a reversible Markov process with voltage-dependent rate constants associated with each respective directional transition. While recovery reconstructed with the Hodgkin-Huxley formalism included a supernormal period, the PC scheme did not. The present analysis showed that the occurrence and degree of supernormality with the PC scheme was determined by the relative speed of the transitions within the closed loop of the kinetic scheme; supernormality was promoted by speeding these kinetics. The analysis also showed that concurrent with supernormality, the faster loop kinetics caused (1) an elevation in the C(1) --> C(2) transitions, and (2) a reduction in the I(4) --> I(5) transitions. Thus, macroscopic functionality in information processing could be expressed in terms of probabilistic interstate transitions among a population of NaCh molecules.

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A simple modification of the Hodgkin and Huxley equations explains type 3 excitability in squid giant axons

Cited by 71 publications

References 39 publications

Multimodal Transition and Excitability of a Neural Oscillator

Multimodal Transition and Excitability of a Neural Oscillator

Switching neuronal state: optimal stimuli revealed using a stochastically-seeded gradient algorithm

Probability Distributions of Markovian Sodium Channel States During Propagating Axonal Impulses With or Without Recovery Supernormality

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