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Published online by Cambridge University Press: 05 June 2025
Large temporal fluctuations or oscillations in cellular states are widely observed in biological systems, for instance, in neural firing, circadian rhythms, and collective motion of amoebae. These phenomena arise from the interplay between positive and negative feedback mechanisms, as discussed in previous chapters. In this chapter, we focus on such dynamic changes in cellular states. Using trajectories of oscillatory dynamics in phase planes such as the Brusselator, we provide detailed explanations of conditions for oscillation through the use of nullcline and Jacobian matrix analyses. We confirm the existence of two mechanisms: the activator-inhibitor system and the substrate-depletion system. Furthermore, we extensively introduce the Hodgkin–Huxley equations concerning membrane potential and excitability, which represent a significant milestone in the fields of biophysics, theoretical biology, and electrophysiology. Through quantitative comparison with experimental data, we elucidate the mechanisms underlying its dynamics, which are explained by the reduction of variables leading to the FitzHugh–Nagumo equations.
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