The classic Hodgkin-Huxley model is widely used for understanding the electrophysiological dynamics of a single neuron. While applying a constant current to the system results in a single voltage spike, it is possible to produce more interesting dynamics by applying time-varying currents, which may not be experimentally measurable. The aim of this work is to estimate time-varying applied currents of different deterministic forms given noisy voltage data. In particular, we utilize an augmented ensemble Kalman filter with parameter tracking to estimate four different deterministic applied currents, analyzing how the model dynamics change in each case. We test the efficiency of the parameter tracking algorithm in this setting by exploring the effects of changing the standard deviation of the parameter drift and the frequency of data available on the resulting time-varying applied current estimates and related uncertainty.

In this paper we introduce a generalization of the inertial manifold, which we call an inertial function. This is a smooth function whose graph is invariant under the dynamics, contains the global attractor, and is exponentially attracting for almost all initial conditions. The construction of this inertial function is based on the Lyapunov exponents of the dynamical system and on properties of the absorbing set.

In this paper we investigate the effects of filtering a chaotic time series with a linear IIR filter. Using the Kaplan and Yorke conjecture it has been argued that such filtering can result in an increase in information dimension. Here we show that the filter dynamics induce an extended dynamical system and that this system possesses a globally attracting invariant graph over the base system. Using this framework we obtain sufficient conditions on the filter dynamics that guarantee that the dimension remains unchanged.

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