Human Brain Networks: Spiking Neuron Models, Multistability, Synchronization, Thermodynamics, Maximum Entropy Production, and Anesthetic Cascade Mechanisms

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.

[1]  Shankar Sastry,et al.  A calculus for computing Filippov's differential inclusion with application to the variable structure control of robot manipulators , 1986, 1986 25th IEEE Conference on Decision and Control.

[2]  L M Riscalla.  An electromagnetic field theory of consciousness. , 1974, The Journal of the American Society of Psychosomatic Dentistry and Medicine.

[3]  Bruce J. West,et al.  Chaos and fractals in human physiology. , 1990, Scientific American.

[4]  Bastian Goldlücke,et al.  Variational Analysis , 2014, Computer Vision, A Reference Guide.

[5]  Sompolinsky,et al.  Spin-glass models of neural networks. , 1985, Physical review. A, General physics.

[6]  Wassim M. Haddad,et al.  Time-reversal symmetry, Poincaré recurrence, irreversibility, and the entropic arrow of time : From mechanics to system thermodynamics , 2008 .

[7]  Wassim M. Haddad,et al.  Reversibility and PoincarÉ Recurrence in Linear Dynamical Systems , 2006, IEEE Transactions on Automatic Control.

[8]  Wulfram Gerstner,et al.  SPIKING NEURON MODELS Single Neurons , Populations , Plasticity , 2002 .

[9]  K. Kuizenga,et al.  Biphasic EEG changes in relation to loss of consciousness during induction with thiopental, propofol, etomidate, midazolam or sevoflurane. , 2001, British journal of anaesthesia.

[10]  Heidi E Kirsch,et al.  The Howling Cortex: Seizures and General Anesthetic Drugs , 2008, Anesthesia and analgesia.

[11]  Andrew J. E. Seely,et al.  Fractal variability: an emergent property of complex dissipative systems. , 2012, Chaos.

[12]  Rod Swenson,et al.  Emergent attractors and the law of maximum entropy production: Foundations to a theory of general evolution , 1989 .

[13]  Xuerong Mao,et al.  Exponential stability of stochastic delay interval systems with Markovian switching , 2002, IEEE Trans. Autom. Control..

[14]  Holger Fröhlich,et al.  Modeling ERBB receptor-regulated G1/S transition to find novel targets for de novo trastuzumab resistance , 2009, BMC Systems Biology.

[15]  G. Buzsáki.  Rhythms of the brain , 2006 .

[16]  Wassim M. Haddad,et al.  A Unification between Dynamical System Theory and Thermodynamics Involving an Energy, Mass, and Entropy State Space Formalism , 2013, Entropy.

[17]  Eric Shea-Brown,et al.  On the Phase Reduction and Response Dynamics of Neural Oscillator Populations , 2004, Neural Computation.

[18]  A. Hutt,et al.  Effects of the anesthetic agent propofol on neural populations , 2010, Cognitive Neurodynamics.

[19]  Leslie S. Prichep,et al.  The Anesthetic Cascade: A Theory of How Anesthesia Suppresses Consciousness , 2005, Anesthesiology.

[20]  E. Overton.  Studien über die Narkose : zugleich ein Beitrag zur allgemeinen Pharmakologie , 1901 .

[21]  E. Eger,et al.  A Comparative Study of Halothane and Halopropane Anesthesia: Including Method for Determining Equipotency , 1963, Anesthesiology.

[22]  Peter Dayan,et al.  Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems , 2001 .

[23]  D. Amit,et al.  Model of global spontaneous activity and local structured activity during delay periods in the cerebral cortex. , 1997, Cerebral cortex.

[24]  E. Ryan.  An Integral Invariance Principle for Differential Inclusions with Applications in Adaptive Control , 1998 .

[25]  Wassim M. Haddad,et al.  Thermodynamics, A Dynamical Systems Approach , 2010 .

[26]  O. Hájek,et al.  Local Semi-Dynamical Systems , 1969 .

[27]  Ilya Prigogine,et al.  From Being To Becoming , 1980 .

[28]  Wassim M. Haddad,et al.  Complexity, robustness, self-organization, swarms, and system thermodynamics☆ , 2009 .

[29]  B. Orser,et al.  Inhaled Anesthetics and Immobility: Mechanisms, Mysteries, and Minimum Alveolar Anesthetic Concentration , 2003, Anesthesia and analgesia.

[30]  J. Leth,et al.  On formalism and stability of switched systems , 2012, Journal of Control Theory and Applications.

[31]  A. Hodgkin,et al.  A quantitative description of membrane current and its application to conduction and excitation in nerve , 1952, The Journal of physiology.

[32]  Jianghai Hu,et al.  Semistability of switched linear systems with applications to distributed sensor networks: A generating function approach , 2011, IEEE Conference on Decision and Control and European Control Conference.

[33]  P. Macklem,et al.  Towards a Definition of Life , 2010, Perspectives in biology and medicine.

[34]  J. Hale,et al.  Ordinary Differential Equations , 2019, Fundamentals of Numerical Mathematics for Physicists and Engineers.

[35]  Sanjay P. Bhat,et al.  Semistability, Finite-Time Stability, Differential Inclusions, and Discontinuous Dynamical Systems Having a Continuum of Equilibria , 2009, IEEE Transactions on Automatic Control.

[36]  Rafael Obaya,et al.  Neutral Functional Differential Equations with Applications to Compartmental Systems , 2008, SIAM J. Math. Anal..

[37]  Wassim M. Haddad,et al.  Temporal Asymmetry, Entropic Irreversibility, and Finite-Time Thermodynamics: From Parmenides-Einstein Time-Reversal Symmetry to the Heraclitan Entropic Arrow of Time , 2012, Entropy.

[38]  Michael C. Mackey,et al.  Propagating fronts, chaos and multistability in a cell replication model. , 1996, Chaos.

[39]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[40]  T. Bullock,et al.  Temporal fluctuations in coherence of brain waves. , 1995, Proceedings of the National Academy of Sciences of the United States of America.

[41]  Richard H. Lyon.  Statistical energy analysis of dynamical systems : theory and applications , 2003 .

[42]  Wassim M. Haddad,et al.  Multistability, bifurcations, and biological neural networks: A synaptic drive firing model for cerebral cortex transition in the induction of general anesthesia , 2011 .

[43]  F W Sharbrough,et al.  Anterior shift of the dominant EEG rhytham during anesthesia in the Java monkey: correlation with anesthetic potency. , 1977, Anesthesiology.

[44]  George A Mashour,et al.  Consciousness Unbound: Toward a Paradigm of General Anesthesia , 2004, Anesthesiology.

[45]  David Terman,et al.  Mathematical foundations of neuroscience , 2010 .

[46]  S. Strogatz.  From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators , 2000 .

[47]  D. Williams.  STOCHASTIC DIFFERENTIAL EQUATIONS: THEORY AND APPLICATIONS , 1976 .

[48]  Qing Hui.  Optimal semistable control for continuous-time linear systems , 2011, Syst. Control. Lett..

[49]  L. Chua,et al.  Synchronization in an array of linearly coupled dynamical systems , 1995 .

[50]  W. T. Thach,et al.  In: Principles of neural science , 2000 .

[51]  T G Buchman,et al.  Uncoupling of biological oscillators: a complementary hypothesis concerning the pathogenesis of multiple organ dysfunction syndrome. , 1996, Critical care medicine.

[52]  J. Cowan,et al.  Field-theoretic approach to fluctuation effects in neural networks. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[53]  W. Haddad,et al.  Drug dosing control in clinical pharmacology , 2005, IEEE Control Systems.

[54]  Eduardo Sontag,et al.  Untangling the wires: A strategy to trace functional interactions in signaling and gene networks , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[55]  D. Angeli,et al.  New analysis technique for multistability detection. , 2006, Systems biology.

[56]  Wassim M. Haddad,et al.  A stochastic mean field model for an excitatory and inhibitory synaptic drive cortical neuronal network , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[57]  C. Peng,et al.  What is physiologic complexity and how does it change with aging and disease? , 2002, Neurobiology of Aging.

[58]  George A Mashour,et al.  Consciousness and the 21st century operating room. , 2013, Anesthesiology.

[59]  E. T. Jaynes,et al.  Papers on probability, statistics and statistical physics , 1983 .

[60]  A. Hill,et al.  The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves , 1910 .

[61]  Karl J. Friston,et al.  The Dynamic Brain: From Spiking Neurons to Neural Masses and Cortical Fields , 2008, PLoS Comput. Biol..

[62]  Jaap Vuyk,et al.  Pharmacodynamics of Alfentanil as a Supplement to Propofol or Nitrous Oxide for Lower Abdominal Surgery in Female Patients , 1993, Anesthesiology.

[63]  A. Pérez-Villalba.  Rhythms of the Brain, G. Buzsáki. Oxford University Press, Madison Avenue, New York (2006), Price: GB £42.00, p. 448, ISBN: 0-19-530106-4 , 2008 .

[64]  J. Gibbs.  On the equilibrium of heterogeneous substances , 1878, American Journal of Science and Arts.

[65]  Dennis S. Bernstein,et al.  Nontangency-Based Lyapunov Tests for Convergence and Stability in Systems Having a Continuum of Equilibria , 2003, SIAM J. Control. Optim..

[66]  Bernhard Hemmer,et al.  Simultaneous Electroencephalographic and Functional Magnetic Resonance Imaging Indicate Impaired Cortical Top–Down Processing in Association with Anesthetic-induced Unconsciousness , 2013, Anesthesiology.

[67]  Gregory D. Smith,et al.  The capacity for multistability in small gene regulatory networks , 2009 .

[68]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[69]  Shouchuan Hu.  Differential equations with discontinuous right-hand sides☆ , 1991 .

[70]  A V Herz,et al.  Neural codes: firing rates and beyond. , 1997, Proceedings of the National Academy of Sciences of the United States of America.

[71]  Zdzisław Denkowski,et al.  Set-Valued Analysis , 2021 .

[72]  UnCheol Lee,et al.  Reconfiguration of Network Hub Structure after Propofol-induced Unconsciousness , 2013, Anesthesiology.

[73]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[74]  Jing Zhou,et al.  Stochastic semistability with application to agreement problems over random networks , 2010, Proceedings of the 2010 American Control Conference.

[75]  E. Schrödinger,et al.  What is life? : the physical aspect of the living cell , 1946 .

[76]  Sommers,et al.  Chaos in random neural networks. , 1988, Physical review letters.

[77]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

[78]  David Pinto,et al.  Mathematical neuroscience: from neurons to circuits to systems , 2003, Journal of Physiology-Paris.

[79]  Jinde Cao,et al.  Global Synchronization of Linearly Hybrid Coupled Networks with Time-Varying Delay , 2008, SIAM J. Appl. Dyn. Syst..

[80]  Johannes Bircher,et al.  Towards a Dynamic Definition of Health and Disease , 2005, Medicine, health care, and philosophy.

[81]  J. Sleigh,et al.  Modelling general anaesthesia as a first-order phase transition in the cortex. , 2004, Progress in biophysics and molecular biology.

[82]  J. Cowan,et al.  Excitatory and inhibitory interactions in localized populations of model neurons. , 1972, Biophysical journal.

[83]  Charles M. Gray,et al.  Synchronous oscillations in neuronal systems: Mechanisms and functions , 1994, Journal of Computational Neuroscience.

[84]  I. Rampil.  A Primer for EEG Signal Processing in Anesthesia , 1998, Anesthesiology.

[85]  R. Ash,et al.  Real analysis and probability , 1975 .

[86]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[87]  William Thomson,et al.  On a universal tendency in nature to the dissipation of mechanical energy , 2007 .

[88]  Nicolas Brunel,et al.  Fast Global Oscillations in Networks of Integrate-and-Fire Neurons with Low Firing Rates , 1999, Neural Computation.

[89]  J. P. Lasalle.  Some Extensions of Liapunov's Second Method , 1960 .

[90]  Riscalla Lm.  An electromagnetic field theory of consciousness. , 1974 .

[91]  Vimal Singh,et al.  Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions , 1979, IEEE Transactions on Systems, Man, and Cybernetics.

[92]  W. Gerstner,et al.  Time structure of the activity in neural network models. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[93]  David Angeli,et al.  Multistability in Systems With Counter-Clockwise Input–Output Dynamics , 2007, IEEE Transactions on Automatic Control.

[94]  Francesco Bullo,et al.  Coordination and Geometric Optimization via Distributed Dynamical Systems , 2003, SIAM J. Control. Optim..

[95]  W. R. Lieb,et al.  Molecular and cellular mechanisms of general anaesthesia , 1994, Nature.

[96]  William Thomson,et al.  2. On a Universal Tendency in Nature to the Dissipation of Mechanical Energy. , 1857 .

[97]  N. Franks,et al.  General anesthesia and ascending arousal pathways. , 2009, Anesthesiology.

[98]  M. Rietkerk,et al.  Self-Organized Patchiness and Catastrophic Shifts in Ecosystems , 2004, Science.

[99]  Edwin T. Jaynes.  Prior Probabilities , 2010, Encyclopedia of Machine Learning.

[100]  S. Amari,et al.  A Mathematical Foundation for Statistical Neurodynamics , 1977 .

[101]  Antonio Loría,et al.  Integral Characterizations of Uniform Asymptotic and Exponential Stability with Applications , 2002, Math. Control. Signals Syst..

[102]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[103]  H. Kamaya,et al.  Molecular Mechanisms of Anesthesia , 1984, The Keio journal of medicine.

[104]  G. P. Szegö,et al.  Stability theory of dynamical systems , 1970 .

[105]  B. Ripley,et al.  E. T. Jaynes: Papers on Probability, Statistics and Statistical Physics , 1983 .

[106]  W. Haddad,et al.  DISCONTINUOUS RIGHT-HAND SIDES : FILIPPOV SOLUTIONS , NONSMOOTH STABILITY AND DISSIPATIVITY THEORY , AND OPTIMAL DISCONTINUOUS FEEDBACK CONTROL , 2014 .

[107]  A. Bacciotti,et al.  Stability and Stabilization of Discontinuous Systems and Nonsmooth Lyapunov Functions , 1999 .

[108]  Sanjay P. Bhat,et al.  Finite-Time Semistability and Consensus for Nonlinear Dynamical Networks , 2008, IEEE Transactions on Automatic Control.

[109]  D. Cafiso,et al.  Contrasting membrane localization and behavior of halogenated cyclobutanes that follow or violate the Meyer-Overton hypothesis of general anesthetic potency. , 1997, Biophysical journal.

[110]  Jean-Marie Aerts,et al.  From data patterns to mechanistic models in acute critical illness. , 2014, Journal of critical care.

[111]  E. Jaynes.  Information Theory and Statistical Mechanics , 1957 .

[112]  Wassim M. Haddad,et al.  Complexity, robustness, self-organization, swarms, and system thermodynamics , 2007, 2007 46th IEEE Conference on Decision and Control.

[113]  W. Haddad,et al.  Nonnegative and Compartmental Dynamical Systems , 2010 .

[114]  T. Narahashi,et al.  Effects of Halothane and Propofol on Excitatory and Inhibitory Synaptic Transmission in Rat Cortical Neurons , 2003, Journal of Pharmacology and Experimental Therapeutics.

[115]  K. Miller,et al.  Mechanisms of actions of inhaled anesthetics. , 2003, The New England journal of medicine.

[116]  Wassim M. Haddad,et al.  Multistability, bifurcations, and biological neural networks: A synaptic drive firing model for cerebral cortex transition in the induction of general anesthesia , 2011, IEEE Conference on Decision and Control and European Control Conference.

[117]  H. Meyer.  Zur Theorie der Alkoholnarkose , 1899, Archiv für experimentelle Pathologie und Pharmakologie.

[118]  Shu-Kun Lin,et al.  Modern Thermodynamics: From Heat Engines to Dissipative Structures , 1999, Entropy.

[119]  W. Haddad,et al.  Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach , 2008 .

[120]  Stuart R. Hameroff,et al.  The Entwined Mysteries of Anesthesia and Consciousness: Is There a Common Underlying Mechanism? , 2006, Anesthesiology.