Theoretical electroencephalogram stationary spectrum for a white-noise-driven cortex: evidence for a general anesthetic-induced phase transition.

We present a model for the dynamics of a cerebral cortex in which inputs to neuronal assemblies are treated as random Gaussian fluctuations about a mean value. We incorporate the effect of general anesthetic agents on the cortex as a modulation of the inhibitory neurotransmitter rate constant. Stochastic differential equations are derived for the state variable h(e), the average excitatory soma potential, coherent fluctuations of which are believed to be the source of scalp-measured electroencephalogram (EEG) signals. Using this stochastic approach we derive a stationary (long-time limit) fluctuation spectrum for h(e). The model predicts that there will be three distinct stationary (equilibrium) regimes for cortical activity. In region I ("coma"), corresponding to a strong inhibitory anesthetic effect, h(e) is single valued, large, and negative, so that neuronal firing rates are suppressed. In region II for a zero or small anesthetic effect, h(e) can take on three values, two of which are stable; we label the stable solutions as "active" (enhanced firing) and "quiescent" (suppressed firing). For region III, corresponding to negative anesthetic (i.e., analeptic) effect, h(e) again becomes single valued, but is now small and negative, resulting in strongly elevated firing rates ("seizure"). If we identify region II as associated with the conscious state of the cortex, then the model predicts that there will be a rapid transit between the active-conscious and comatose unconscious states at a critical value of anesthetic concentration, suggesting the existence of phase transitions in the cortex. The low-frequency spectral power in the h(e) signal should increase strongly during the initial stage of anesthesia induction, before collapsing to much lower values after the transition into comatose-unconsciousness. These qualitative predictions are consistent with clinical measurements by Bührer et al. [Anaesthesiology 77, 226 (1992)], MacIver et al. [ibid. 84, 1411 (1996)], and Kuizenga et al. [Br. J. Anaesthesia 80, 725 (1998)]. This strong increase in EEG spectral power in the vicinity of the critical point is similar to the divergences observed during thermodynamic phase transitions. We show that the divergence in low-frequency power in our model is a natural consequence of the existence of turning points in the trajectory of stationary states for the cortex.

[1]  Heinz Zemanek,et al.  Kybernetik , 1964, Elektron. Rechenanlagen.

[2]  Handbuch der Physik: Vol. XLIX/1 Editor: S. Flügge. Geophysics III/1, Group Editor, J. Bartels. 315 pages, diagrams, illustr. 612 × 934 in Berlin, New York, Springer-Verlag, 1966. Price, 120 DM (cloth), [96 DM (subscr.) $30.00 (approx.)] , 1968 .

[3]  R. Elul.  The genesis of the EEG. , 1971, International review of neurobiology.

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

[5]  Donald O. Walter,et al.  Mass action in the nervous system , 1975 .

[6]  K. Kaplan.  H. Haken, Synergetics. An Introduction. Nonequilibrium Phase Transitions and Self-Organization in Physics, Chemistry, and Biology (2nd Edition). XI + 355 S., 152 Abb. Berlin—Heidelberg—New York 1978. Springer-Verlag. DM 66,00 , 1980 .

[7]  A. J. Hermans,et al.  A model of the spatial-temporal characteristics of the alpha rhythm , 1982 .

[8]  J. Fermaglich.  Electric Fields of the Brain: The Neurophysics of EEG , 1982 .

[9]  S. Paul,et al.  Convulsant potencies of tetrazoles are highly correlated with actions on GABA/benzodiazepine/picrotoxin receptor complexes in brain. , 1984, Life sciences.

[10]  H. Haken.  Self-Organization in Physics , 1985 .

[11]  K. Vahala.  Handbook of stochastic methods for physics, chemistry and the natural sciences , 1986, IEEE Journal of Quantum Electronics.

[12]  E. G. Jones.  Cerebral Cortex , 1987, Cerebral Cortex.

[13]  I. Langmoen,et al.  Isoflurane hyperpolarizes neurones in rat and human cerebral cortex. , 1987, Acta physiologica Scandinavica.

[14]  H. Tuckwell.  Introduction to Theoretical Neurobiology: Linear Cable Theory and Dendritic Structure , 1988 .

[15]  Henry C. Tuckwell,et al.  Introduction to theoretical neurobiology , 1988 .

[16]  Daniel J. Amit,et al.  Modeling brain function: the world of attractor neural networks, 1st Edition , 1989 .

[17]  S L Shafer,et al.  Thiopental pharmacodynamics. I. Defining the pseudo-steady-state serum concentration-EEG effect relationship. , 1992, Anesthesiology.

[18]  N L Harrison,et al.  Effects of volatile anesthetics on the kinetics of inhibitory postsynaptic currents in cultured rat hippocampal neurons. , 1993, Journal of neurophysiology.

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

[20]  H. Haken,et al.  Field Theory of Electromagnetic Brain Activity. , 1996, Physical review letters.

[21]  David Ferster,et al.  Is Neural Noise Just a Nuisance? , 1996, Science.

[22]  James J. Wright,et al.  Dynamics of the brain at global and microscopic scales: Neural networks and the EEG , 1996, Behavioral and Brain Sciences.

[23]  B. H. Bland,et al.  Thiopental Uncouples Hippocampal and Cortical Synchronized Electroencephalograpbic Activity , 1996, Anesthesiology.

[24]  C. Koch,et al.  A brief history of time (constants). , 1996, Cerebral cortex.

[25]  James J. Wright,et al.  Propagation and stability of waves of electrical activity in the cerebral cortex , 1997 .

[26]  S. Roth,et al.  Pharmacodynamics of thiopentone: nocifensive reflex threshold changes correlate with hippocampal electroencephalography. , 1997, British journal of anaesthesia.

[27]  B. Antkowiak,et al.  Cellular mechanisms of gamma rhythms in rat neocortical brain slices probed by the volatile anaesthetic isoflurane , 1997, Neuroscience Letters.

[28]  H. Haken,et al.  A derivation of a macroscopic field theory of the brain from the quasi-microscopic neural dynamics , 1997 .

[29]  K. Kuizenga,et al.  Quantitative electroencephalographic analysis of the biphasic concentration-effect relationship of propofol in surgical patients during extradural analgesia. , 1998, British journal of anaesthesia.

[30]  Peter N. Robinson,et al.  STEADY STATES AND GLOBAL DYNAMICS OF ELECTRICAL ACTIVITY IN THE CEREBRAL CORTEX , 1998 .

[31]  B. Antkowiak,et al.  Effects of Small Concentrations of Volatile Anesthetics on Action Potential Firing of Neocortical Neurons In Vitro , 1998, Anesthesiology.

[32]  David T. J. Liley,et al.  A continuum theory of electro-cortical activity , 1999, Neurocomputing.

[33]  H. Haken,et al.  Impacts of noise on a field theoretical model of the human brain , 1999 .