The recovery of (weak) stimuli encoded with a population of Hodgkin-Huxley neurons is investigated. In the absence of a stimulus, the Hodgkin-Huxley neurons are assumed to be tonically spiking. The methodology employed calls for 1) finding an input-output (I/O) equivalent description of the Hodgkin-Huxley neuron and 2) devising a recovery algorithm for stimuli encoded with the I/O equivalent neuron(s). A Hodgkin-Huxley neuron with multiplicative coupling is I/O equivalent with an Integrate-and-Fire neuron with a variable threshold sequence. For bandlimited stimuli a perfect recovery of the stimulus can be achieved provided that a Nyquist-type rate condition is satisfied. A Hodgkin-Huxley neuron with additive coupling and deterministic conductances is first-order I/O equivalent with a Project-Integrate-and-Fire neuron that integrates a projection of the stimulus on the phase response curve. The stimulus recovery is formulated as a spline interpolation problem in the space of finite length bounded energy signals. A Hodgkin-Huxley neuron with additive coupling and stochastic conductances is shown to be first-order I/O equivalent with a Project-Integrate-and-Fire neuron with random thresholds. For stimuli modeled as elements of Sobolev spaces the reconstruction algorithm minimizes a regularized quadratic optimality criterion. Finally, all previous recovery results of stimuli encoded with Hodgkin-Huxley neurons with multiplicative and additive coupling, and deterministic and stochastic conductances are extended to stimuli encoded with a population of Hodgkin-Huxley neurons.
Keywords: Hodgkin–Huxley neurons; input–output (I/O) equivalence; neural encoding; population encoding; reproducing Kernel Hilbert spaces; splines; stimulus reconstruction.