Abstract
Advances in neuroscience have been closely linked to mathematical modeling beginning with the integrate-and-fire model of Lapicque and proceeding through the modeling of the action potential by Hodgkin and Huxley to the current era. The fundamental building block of the central nervous system, the neuron, may be thought of as a dynamic element that is “excitable”, and can generate a pulse or spike whenever the electrochemical potential across the cell membrane of the neuron exceeds a threshold. A key application of nonlinear dynamical systems theory to the neurosciences is to study phenomena of the central nervous system that exhibit nearly discontinuous transitions between macroscopic states. A very challenging and clinically important problem exhibiting this phenomenon is the induction of general anesthesia. In any specific patient, the transition from consciousness to unconsciousness as the concentration of anesthetic drugs increases is very sharp, resembling a thermodynamic phase transition. This paper focuses on multistability theory for continuous and discontinuous dynamical systems having a set of multiple isolated equilibria and/or a continuum of equilibria. Multistability is the property whereby the solutions of a dynamical system can alternate between two or more mutually exclusive Lyapunov stable and convergent equilibrium states under asymptotically slowly changing inputs or system parameters. In this paper, we extend the theory of multistability to continuous, discontinuous, and stochastic nonlinear dynamical systems. In particular, Lyapunov-based tests for multistability and synchronization of dynamical systems with continuously differentiable and absolutely continuous flows are established. The results are then applied to excitatory and inhibitory biological neuronal networks to explain the underlying mechanism of action for anesthesia and consciousness from a multistable dynamical system perspective, thereby providing a theoretical foundation for general anesthesia using the network properties of the brain. Finally, we present some key emergent properties from the fields of thermodynamics and electromagnetic field theory to qualitatively explain the underlying neuronal mechanisms of action for anesthesia and consciousness.
Highlights
Advances in neuroscience have been closely linked to mathematical modeling beginning with the integrate-and-fire model of Lapicque [1] and proceeding through the modeling of the action potential by Hodgkin and Huxley [2] to the current era of mathematical neuroscience; see [3,4] and the numerous references therein
This concept is certainly applicable for the administration of general inhalational anesthetics, where the potency of the drug is defined by the minimum alveolar concentration (MAC) of the drug needed to prevent a response to noxious stimuli in 50% of administrations [11]
Thermodynamics, Neuroscience, Consciousness, and the Entropic Arrow of Time. In this and the section, we present some qualitative insights from the fields of thermodynamics and electromagnetic field theory that can potentially be useful in developing mechanistic models [100] that can explain the underlying mechanisms of action for general anesthesia and consciousness
Summary
Advances in neuroscience have been closely linked to mathematical modeling beginning with the integrate-and-fire model of Lapicque [1] and proceeding through the modeling of the action potential by Hodgkin and Huxley [2] to the current era of mathematical neuroscience; see [3,4] and the numerous references therein. An interesting example of how changes in the postsynaptic potential may be applied to the analysis of the induction of anesthesia is the view of anesthesia as a phase transition proposed by Steyn-Ross et al (see [15] and the references therein) While their analysis was highly informative, in this paper we use a dynamical system theory framework in terms of neuronal firing rates, using the concepts of network thermodynamics [32] and multistability, for explaining the mechanisms of action for general anesthesia. This facilitates a focus on the network properties of the central nervous system.
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