Neural field theory with variance dynamics

Previous neural field models have mostly been concerned with prediction of mean neural activity and with second order quantities such as its variance, but without feedback of second order quantities on the dynamics. Here the effects of feedback of the variance on the steady states and adiabatic dynamics of neural systems are calculated using linear neural field theory to estimate the neural voltage variance, then including this quantity in the total variance parameter of the nonlinear firing rate-voltage response function, and thus into determination of the fixed points and the variance itself. The general results further clarify the limits of validity of approaches with and without inclusion of variance dynamics. Specific applications show that stability against a saddle-node bifurcation is reduced in a purely cortical system, but can be either increased or decreased in the corticothalamic case, depending on the initial state. Estimates of critical variance scalings near saddle-node bifurcation are also found, including physiologically based normalizations and new scalings for mean firing rate and the position of the bifurcation.

[1]  John R. Terry,et al.  A unifying explanation of primary generalized seizures through nonlinear brain modeling and bifurcation analysis. , 2006, Cerebral cortex.

[2]  Karl J. Friston,et al.  Population dynamics: Variance and the sigmoid activation function , 2008, NeuroImage.

[3]  Prof. Dr. Dr. Valentino Braitenberg,et al.  Cortex: Statistics and Geometry of Neuronal Connectivity , 1998, Springer Berlin Heidelberg.

[4]  Jian-Young Wu,et al.  Compression and Reflection of Visually Evoked Cortical Waves , 2007, Neuron.

[5]  Donald L Rowe,et al.  Estimation of neurophysiological parameters from the waking EEG using a biophysical model of brain dynamics. , 2004, Journal of theoretical biology.

[6]  P. Robinson,et al.  Modal analysis of corticothalamic dynamics, electroencephalographic spectra, and evoked potentials. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  Michael A. Buice,et al.  Systematic Fluctuation Expansion for Neural Network Activity Equations , 2009, Neural Computation.

[8]  P. Robinson Propagator theory of brain dynamics. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[10]  J. Sleigh,et al.  Toward a theory of the general-anesthetic-induced phase transition of the cerebral cortex. II. Numerical simulations, spectral entropy, and correlation times. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  P. Robinson,et al.  Dynamics of large-scale brain activity in normal arousal states and epileptic seizures. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  David B. Grayden,et al.  An Analytical Model for the ‘Large, Fluctuating Synaptic Conductance State’ Typical of Neocortical Neurons In Vivo , 2004, Journal of Computational Neuroscience.

[13]  P A Robinson,et al.  Wave-number spectrum of electroencephalographic signals. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  P. Nunez,et al.  Electric fields of the brain , 1981 .

[15]  Paul C Bressloff,et al.  Bloch waves, periodic feature maps, and cortical pattern formation. , 2002, Physical review letters.

[16]  J. Cowan,et al.  A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue , 1973, Kybernetik.

[17]  Steven J. Schiff,et al.  Dynamical evolution of spatiotemporal patterns in mammalian middle cortex. , 2007 .

[18]  J. Parra,et al.  Epileptic Transitions: Model Predictions and Experimental Validation , 2005, Journal of clinical neurophysiology : official publication of the American Electroencephalographic Society.

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

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

[21]  I A Lubashevsky,et al.  Fast heat propagation in living tissue caused by branching artery network. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  N. Hatsopoulos,et al.  Propagating waves mediate information transfer in the motor cortex , 2006, Nature Neuroscience.

[23]  P A Robinson,et al.  Neural rate equations for bursting dynamics derived from conductance-based equations. , 2008, Journal of theoretical biology.

[24]  Fabrice Wendling,et al.  Some Insights Into Computational Models of (Patho)physiological Brain Activity , 2006, Proceedings of the IEEE.

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

[26]  D A Steyn-Ross,et al.  Toward a theory of the general-anesthetic-induced phase transition of the cerebral cortex. I. A thermodynamics analogy. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  D. Liley,et al.  Theoretical electroencephalogram stationary spectrum for a white-noise-driven cortex: evidence for a general anesthetic-induced phase transition. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

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

[29]  P. Nunez Wavelike Properties of the Alpha Rhythm , 1974 .

[30]  J. Cowan,et al.  Statistical mechanics of the neocortex. , 2009, Progress in biophysics and molecular biology.

[31]  L. Garey Cortex: Statistics and Geometry of Neuronal Connectivity, 2nd edn. By V. BRAITENBERG and A. SCHÜZ. (Pp. xiii+249; 90 figures; ISBN 3 540 63816 4). Berlin: Springer. 1998. , 1999 .

[32]  Moira L Steyn-Ross,et al.  Proposed mechanism for learning and memory erasure in a white-noise-driven sleeping cortex. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[33]  José F Fontanari,et al.  Preservation of information in a prebiotic package model. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[34]  Paul C. Bressloff,et al.  Stochastic Neural Field Theory and the System-Size Expansion , 2009, SIAM J. Appl. Math..

[35]  P A Robinson,et al.  Estimation of multiscale neurophysiologic parameters by electroencephalographic means , 2004, Human brain mapping.

[36]  P. Robinson,et al.  Compact dynamical model of brain activity. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[37]  Karl J. Friston,et al.  A dynamic causal model study of neuronal population dynamics , 2010, NeuroImage.

[38]  D. Liley,et al.  Understanding the Transition to Seizure by Modeling the Epileptiform Activity of General Anesthetic Agents , 2005, Journal of clinical neurophysiology : official publication of the American Electroencephalographic Society.

[39]  D. Liley,et al.  Modeling the effects of anesthesia on the electroencephalogram. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[40]  P. Nunez,et al.  Neocortical Dynamics and Human EEG Rhythms , 1995 .

[41]  J. Cowan,et al.  SO3 symmetry breaking mechanism for orientation and spatial frequency tuning in the visual cortex. , 2002, Physical review letters.

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

[43]  Peter A. Robinson,et al.  Unified neurophysical model of EEG spectra and evoked potentials , 2002, Biological Cybernetics.

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

[45]  F. H. Lopes da Silva,et al.  Model of brain rhythmic activity , 1974, Kybernetik.

[46]  P A Robinson,et al.  Wave-number spectrum of electrocorticographic signals. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[47]  J. Sleigh,et al.  The Sleep Cycle Modelled as a Cortical Phase Transition , 2005, Journal of biological physics.

[48]  P. Robinson,et al.  Prediction of electroencephalographic spectra from neurophysiology. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.