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Chaotic states in a random world: Relationship between the nonlinear differential equations of excitability and the stochastic properties of ion channels


Excitable membranes allow cells to generate and propagate electrical signals. In the nervous system these signals transmit information, in muscle they trigger contraction, and in heart they regulate spontaneous beating. A central question in excitability theory concerns the relationship between the aggregate properties of membranes (marcoscopic) and the properties of channels in the membranes (mircoscopic). Hodgkin and Huxley (1952) laid the foundations of membrane excitability, and Neher and Sakmann (1976) developed techniques to study individual channels. This article focuses on the relationship between the macroscopic domain, in which non-linear differential equations determine the electrical properties of cells, and the microscopic domain, in which the probabilistic nature of channels establishes the pattern of activity. Using nerve cell membranes as an example, we examine how information in one domain predicts behavior in the other. We conclude that the probabilistic nature of channels generates virtually all macroscopic electrical properties, including resting potentials, action potentials, spontaneous firing, and chaos.

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DeFelice, L.J., Isaac, A. Chaotic states in a random world: Relationship between the nonlinear differential equations of excitability and the stochastic properties of ion channels. J Stat Phys 70, 339–354 (1993). https://doi.org/10.1007/BF01053972

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Key words

  • probabilistic ion channels
  • resting potential
  • action potential
  • spontaneous firing
  • chaos