| Sensory neurons respond to stimulatory signals with periodic firings of the membrane potential. The frequency of firing is often proportional to the intensity of stimulus, and one says that the stimulus is frequency encoded. Frequency encoding has significant theoretical and application value in studying neurobiology. Even since the first publication of Hodgkin-Huxley landmark work, we know that a neuron can be described by a system of differential equations. Due to the complexity, it’s not a good system for studying frequency encoding. In this thesis, we simplify the original Hodgkin-Huxley system into a2-equation system. Here we develop a theory of frequency encoding in systems described by differential equations with decoupling time scales. The main contributions include:1. Based on the original Hodgkin-Huxley system, we simplify it into a2-equation system. We remove the fast dynamics of the original system, and combine the slow dynamics together. The reduced system has similar properties as the original HH system. Compare with FitzHugh-Nagumo model, the reduced system is a more loyal simplification to the original system.2. We develop a theory of frequency encoding in systems described by differential equations with decoupling time scales, i.e. the singularly perturbed systems. Three distinct modes of frequency encoding, namely, the absolute mode, the exact mode, and the inexact mode, are identified. It is shown that frequency encoding through injecting current is of the inexact mode and it has limited range of frequency variations.3. Theoretical studies show how different modes of frequency encoding arise as a single parameter of the system is varied, including system with same scaling factors and distinct scaling factors for the slow dynamics. Phosphorylation of the K channel selectively down-regulations the kinetic rate of K channel and can greatly expand the frequency domain of this encoding. Additionally, the axon can encode the analog signal of a time-varying (hormone). |
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