Chaos in neurobiology

Deterministic mathematical models of neural systems can give rise to complex aperiodic (`chaotic') dynamics in the absence of stochastic fluctuations (`noise') in the variables or parameters of the model or in the inputs to the system. The authors show that chaotic dynamics are expected in nonlinear feedback systems possessing time delays such as are found in recurrent inhibition and from the periodic forcing of neural oscillators. The implications of the possible occurrence of chaotic dynamics for experimental work and mathematical modeling of normal and abnormal functions neurophysiology are mentioned.

[1]  C. Sparrow,et al.  Bifurcation and chaotic behaviour in simple feedback systems. , 1980, Journal of theoretical biology.

[2]  Jose Antonio Coarasa Perez,et al.  Evidence for universal chaotic behavior of a driven nonlinear oscillator , 1982 .

[3]  L. Harmon,et al.  Neural modeling. , 1966, Physiological reviews.

[4]  L. Glass,et al.  Phase locking, period-doubling bifurcations, and irregular dynamics in periodically stimulated cardiac cells. , 1981, Science.

[5]  O. Rössler.  Chaotic Behavior in Simple Reaction Systems , 1976 .

[6]  L. Danziger,et al.  The thyroid-pituitary homeostatic mechanism , 1956 .

[7]  J. Eckmann.  Roads to turbulence in dissipative dynamical systems , 1981 .

[8]  L. Glass,et al.  Unstable dynamics of a periodically driven oscillator in the presence of noise. , 1980, Journal of theoretical biology.

[9]  J. Cronin,et al.  Mathematical aspects of periodic catatonic schizophrenia. , 1977, Bulletin of mathematical biology.

[10]  S. Rush,et al.  A Practical Algorithm for Solving Dynamic Membrane Equations , 1978, IEEE Transactions on Biomedical Engineering.

[11]  R. Pérez,et al.  Bifurcation and chaos in a periodically stimulated cardiac oscillator , 1983 .

[12]  C A Czeisler,et al.  Human sleep: its duration and organization depend on its circadian phase. , 1980, Science.

[13]  M. Feigenbaum.  The universal metric properties of nonlinear transformations , 1979 .

[14]  M C Mackey,et al.  The extinction of slowly evolving dynamical systems , 1980, Journal of mathematical biology.

[15]  C. Hayashi,et al.  Nonlinear oscillations in physical systems , 1987 .

[16]  B. A. Huberman,et al.  Theory of intermittency , 1982 .

[17]  Arkady Pikovsky,et al.  A dynamical model for periodic and chaotic oscillations in the Belousov-Zhabotinsky reaction , 1981 .

[18]  Harry L. Swinney,et al.  BIFURCATIONS TO PERIODIC, QUASIPERIODIC, AND CHAOTIC REGIMES IN ROTATING AND CONVECTING FLUIDS * , 1979 .

[19]  J. Rinzel,et al.  Control of repetitive firing in squid axon membrane as a model for a neuroneoscillator. , 1980, The Journal of physiology.

[20]  U. an der Heiden,et al.  The dynamics of recurrent inhibition , 1984, Journal of mathematical biology.

[21]  K. O. Johnson,et al.  Reconstruction of population response to a vibratory stimulus in quickly adapting mechanoreceptive afferent fiber population innervating glabrous skin of the monkey. , 1974, Journal of neurophysiology.

[22]  L. Glass,et al.  PATHOLOGICAL CONDITIONS RESULTING FROM INSTABILITIES IN PHYSIOLOGICAL CONTROL SYSTEMS * , 1979, Annals of the New York Academy of Sciences.

[23]  Robert M. May,et al.  Simple mathematical models with very complicated dynamics , 1976, Nature.

[24]  D. Prince,et al.  Neurophysiology of epilepsy. , 1978, Annual review of neuroscience.

[25]  R. FitzHugh.  Impulses and Physiological States in Theoretical Models of Nerve Membrane. , 1961, Biophysical journal.

[26]  John Guckenheimer,et al.  Noise in chaotic systems , 1982 .

[27]  Gail A. Carpenter,et al.  Bursting Phenomena in Excitable Membranes , 1979 .

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

[29]  J. Rinzel,et al.  Numerical calculation of stable and unstable periodic solutions to the Hodgkin-Huxley equations , 1980 .

[30]  M C Mackey,et al.  Periodic auto-immune hemolytic anemia: an induced dynamical disease. , 1979, Bulletin of mathematical biology.

[31]  E. Ott.  Strange attractors and chaotic motions of dynamical systems , 1981 .

[32]  H. Walther,et al.  Existence of chaos in control systems with delayed feedback , 1983 .

[33]  N. Macdonald,et al.  Time lag in a model of a biochemical reaction sequence with end product inhibition. , 1977, Journal of theoretical biology.

[34]  J. Shaw.  EEG-brain dynamics E. Basar Elsevier/North-Holland, Amsterdam, 1980, pp. 411 Dfl. 184 , 1982, Biological Psychology.

[35]  S. Fauve,et al.  Period doubling cascade in mercury, a quantitative measurement , 1982 .

[36]  N. Levinson,et al.  A Second Order Differential Equation with Singular Solutions , 1949 .

[37]  B A Huberman,et al.  Chaotic behavior in dopamine neurodynamics. , 1984, Proceedings of the National Academy of Sciences of the United States of America.

[38]  A. Wolf,et al.  One-Dimensional Dynamics in a Multicomponent Chemical Reaction , 1982 .

[39]  R. Pérez,et al.  Fine Structure of Phase Locking , 1982 .

[40]  F. Dudek,et al.  Synchronous neural afterdischarges in rat hippocampal slices without active chemical synapses. , 1982, Science.

[41]  H. Pinsker.  Aplysia bursting neurons as endogenous oscillators. II. Synchronization and entrainment by pulsed inhibitory synaptic input. , 1977, Journal of neurophysiology.

[42]  L Glass,et al.  Phase locking, period doubling bifurcations and chaos in a mathematical model of a periodically driven oscillator: A theory for the entrainment of biological oscillators and the generation of cardiac dysrhythmias , 1982, Journal of mathematical biology.

[43]  E. N. Best,et al.  Null space in the Hodgkin-Huxley Equations. A critical test. , 1979, Biophysical journal.

[44]  Irving B. Elliott.  FLAP, a simplified simulation system for time-share users , 1972 .

[45]  Stein Ps.  Application of the mathematics of coupled oscillator systems to the analysis of the neural control of locomotion. , 1977 .

[46]  M MANZOTTI,et al.  The effect of some respiratory manœuvres on the heart rate , 1958, The Journal of physiology.

[47]  M C Mackey,et al.  Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis. , 1978, Blood.

[48]  G. P. Moore,et al.  PACEMAKER NEURONS: EFFECTS OF REGULARLY SPACED SYNAPTIC INPUT. , 1964, Science.

[49]  Bruce W. Knight,et al.  Dynamics of Encoding in a Population of Neurons , 1972, The Journal of general physiology.

[50]  A. Huxley.  ION MOVEMENTS DURING NERVE ACTIVITY , 1959, Annals of the New York Academy of Sciences.

[51]  P. Andersen.  Organization of Hippocampal Neurons and Their Interconnections , 1975 .

[52]  L. Glass,et al.  Oscillation and chaos in physiological control systems. , 1977, Science.

[53]  Hatsuo Hayashi,et al.  Chaos in the self-sustained oscillation of an excitable biological membrane under sinusoidal stimulation , 1982 .