Abstract
The dynamics of a spiking neuron approaching threshold is investigated in the framework of Markov-chain models describing the random state-transitions of the underlying ion-channel proteins. We characterize subthreshold channel-noise-induced transmembrane potential fluctuations in both type-I (integrator) and type-II (resonator) parametrizations of the classic conductance-based Hodgkin-Huxley equations. As each neuron approaches spiking threshold from below, numerical simulations of stochastic trajectories demonstrate pronounced growth in amplitude simultaneous with decay in frequency of membrane voltage fluctuations induced by ion-channel state transitions. To explore this progression of fluctuation statistics, we approximate the exact Markov treatment with a 12-variable channel-based stochastic differential equation (SDE) and its Ornstein-Uhlenbeck (OU) linearization and show excellent agreement between Markov and SDE numerical simulations. Predictions of the OU theory with respect to membrane potential fluctuation variance, autocorrelation, correlation time, and spectral density are also in agreement and illustrate the close connection between the eigenvalue structure of the associated deterministic bifurcations and the observed behavior of the noisy Markov traces on close approach to threshold for both integrator and resonator point-neuron varieties.
Highlights
As described by Hodgkin-Huxley type models [1], the nonlinear dependence of ion-channel conductance on transmembrane voltage is the basis of action potential generation and provides a link between a neuron’s subthreshold response and suprathreshold dynamics
We find that numerical simulations of the full nonlinear stochastic differential equation (SDE), as well as predictions derived from its Ornstein-Uhlenbeck linearization, accurately capture the subthreshold fluctuation statistics of the detailed Markov-chain model
For the type-I integrator in Fig. 6(a), there is increasing spectral coloration of transmembrane potential fluctuations sharply peaked around ω = 0 as the input current density approaches threshold. This behavior contrasts with the type-II resonator in Fig. 6(b) where we see a resonance peak (∼61 Hz at IDC = 0, ∼85 Hz at 6.0 μA/cm2) whose magnitude diverges as the resonance becomes more narrowly focused for IDC → IDcrCit
Summary
As described by Hodgkin-Huxley type models [1], the nonlinear dependence of ion-channel conductance on transmembrane voltage is the basis of action potential generation and provides a link between a neuron’s subthreshold response and suprathreshold dynamics. In this paper we investigate the influence of ion-channel noise on the subthreshold dynamical behavior of two parametrizations of the Hodgkin-Huxley point-neuron model via numerical simulation of the underlying Markov-chain ion-channel models as the input current threshold for action potential generation is approached from below. We find that numerical simulations of the full nonlinear SDE, as well as predictions derived from its Ornstein-Uhlenbeck linearization, accurately capture the subthreshold fluctuation statistics of the detailed Markov-chain model This progression of models—from Markov, to full SDE, and, to linear SDE—demonstrates the close connection between the observed behavior of the Markov stochastic trajectories and the eigenvalue structure of the associated deterministic bifurcations on approach to threshold
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