Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bojak I, Liley DTJ (2005) Modeling the effects of anesthesia on the electroencephalogram. J Clin Neurophysiol 22: 300
Braitenberg V, Schüz A (1998) Cortex: statistics and geometry of neuronal connectivity 2nd edn. Springer, Berlin
Breakspear M, Roberts JA, Terry JR, Rodrigues S, Mahant N, Robinson PA (2006) A unifying explanation of primary generalized seizures through nonlinear brain modeling and bifurcation analysis. Cereb Cortex 16: 1296
Bressloff PC (2002) Bloch waves, periodic feature maps, and cortical pattern formation. Phys Rev Lett 59: 088101
Bressloff PC (2009) Stochastic neural field theory and the system-size expansion. SIAM J Appl Math 70: 1488
Bressloff PC, Cowan JD (2002) SO (3) symmetry breaking mechanism for orientation and spatial frequency tuning in the visual cortex. Phys Rev Lett 88: 078102
Buice MA, Cowan JD (2007) Field-theoretic approach to fluctuation effects in neural networks. Phys Rev E 75: 051909
Buice MA, Cowan JD (2009) Statistical mechanics of the neocortex. Prog Biophys Molec Biol 99: 53
Buice MA, Cowan JD, Chow CC (2010) Systematic fluctuation expansion for neural network activity equations. Neural Comput 22: 377
Deco G, Jirsa VK, Robinson PA, Breakspear M, Friston K (2008) The dynamic brain: from spiking neurons to neural masses and cortical fields. Publ Lib Sci Comp Biol 4: e1000092
Freeman WJ (1975) Mass action in the nervous system. Academic, New York
Jirsa VK, Haken H (1996) Field theory of electromagnetic brain activity. Phys Rev Lett 77: 960–963
Kim JW, Robinson PA (2007) Compact dynamical model of brain activity. Phys Rev E 75: 031907
Lopes da Silva FH, Hoeks A, Smits H, Zetterberg LH (1974) Model of brain rhythmic activity the alpha-rhythm of the thalamus. Kybernetik 15: 27–37
Marreiros AC, Daunizeau J, Kiebel SJ, Friston KJ (2008) Population dynamics: variance and the sigmoidal activation function. Neuroimage 42: 147
Marreiros AC, Kiebel SJ, Friston KJ (2010) A dynamic causal model study of neuronal population dynamics. Neuroimage 51: 91
Meffin H, Burkitt AN, Grayden DB (2004) An analytical model for the ’large, fluctuating synaptic conductance state’ typical of neocortical neurons in vivo. J Comp Neurosci 16:159–175 and references cited therein
Nunez PL (1974) Wavelike properties of the alpha rhythm. IEEE Trans Biomed Eng 21: 473–482
Nunez PL (1995) Neocortical dynamics and human EEG rhythms. Oxford University Press, Oxford
Nunez PL, Srinivasan R (2006) Electric fields of the brain. Oxford University Press, Oxford
O’Connor SC, Robinson PA (2003) Wave-number spectrum of electrocorticographic signals. Phys Rev E 67: 051912
O’Connor SC, Robinson PA, Chiang AKI (2002) Wave-number spectrum of electroencephalographic signals. Phys Rev E 66: 061905
Rennie CJ, Robinson PA, Wright JJ (2002) Unified neurophysical model of EEG spectra and evoked potentials. Biol Cybern 86: 457–471
Robinson PA (2005) Propagator theory of brain dynamics. Phys Rev E 72: 011904
Robinson PA, Loxley PN, O’Connor SC, Rennie CJ (2000) Modal analysis of corticothalamic dynamics, electroencephalographic spectra, and evoked potentials. Phys Rev E 63: 041909
Robinson PA, Rennie CJ, Rowe DL (2002) Dynamics of large-scale brain activity in normal arousal states and epileptic seizures. Phys Rev E 65: 041924
Robinson PA, Rennie CJ, Rowe DL, O’Connor SC (2004) Estimation of multiscale neurophysiologic parameters by electroencephalographic means. Hum Brain Mapp 23: 53–72
Robinson PA, Rennie CJ, Wright JJ (1997) Propagation and stability of waves of electrical activity in the cerebral cortex. Phys Rev E 56: 826–840
Robinson PA, Rennie CJ, Wright JJ, Bourke PD (1998) Steady states and global dynamics of electrical activity in the cerebral cortex. Phys Rev E 58: 3557
Robinson PA, Rennie CJ, Wright JJ, Bahramali H, Gordon E, Rowe DL (2001) Prediction of electroencephalographic spectra from neurophysiology. Phys Rev E 63: 021903
Robinson PA, Wu H, Kim JW (2007) Neural rate equations for bursting dynamics derived from conductance-based equation. J Theor Biol 250: 663–672
Rowe DL, Robinson PA, Rennie CJ (2004) Estimation of neurophysiological parameters from the waking EEG using a biophysical model of brain dynamics. J Theor Biol 231: 413–433
Rubino D, Robbins KA, Hatsopoulos NG (2006) Propagating waves mediate information transfer in the motor cortex. Nat Neurosci 9: 1549–1557
Schiff SJ, Huang X, Wu JY (2007) Dynamical evolution of spatiotemporal patterns in mammalian middle cortex. Phys Rev Lett 98: 178102
Steyn-Ross DA, Steyn-Ross ML, Sleigh JW, Wilson MT, Gillies IP, Wright JJ (2005) The sleep cycle modelled as a cortical phase transition. J Biol Phys 31: 547–569
Steyn-Ross DA, Steyn-Ross ML, Wilcocks LC, Sleigh JW (2001) Toward a theory of the general-anesthetic-induced phase transition of the cerebral cortex. II. Numerical simulations, spectral entropy, and correlation times. Phys Rev E 64: 011918
Steyn-Ross ML, Steyn-Ross DA, Sleigh JW, Liley DTJ (1999) Theoretical electroencephalogram stationary spectrum for a white-noise-driven cortex: Evidence for a general anesthetic-induced phase transition. Phys Rev E 60: 7299
Steyn-Ross ML, Steyn-Ross DA, Sleigh JW, Wilcocks LC (2001) Toward a theory of the general-anesthetic-induced phase transition of the cerebral cortex. I. A thermodynamics analogy. Phys Rev E 64: 011917
Steyn-Ross ML, Steyn-Ross DA, Sleigh JW, Wilson MT, Wilcocks LC (2005) Proposed mechanism for learning and memory erasure in a white-noise-driven sleeping cortex. Phys Rev E 72: 061910
Steyn-Ross ML, Steyn-Ross DA, Sleigh JW, Wilson MT, Wilcocks LC (2005) Proposed mechanism for learning and memory erasure in a white-noise-driven sleeping cortex. Phys Rev E 72: 051920
Suffczynski P, Lopes da Silva FH, Parra J, Velis D, Kalitzin S (2005) Epileptic transitions: model predictions and experimental validation. J Clin Neurophysiol 22: 288
Suffczynski P, Wendling F, Bellanger JJ, Da Silva FHL (2006) Some insights into computational models of (Patho) physiological brain activity. Proc IEEE 94: 784–804
Wilson HR, Cowan JD (1973) Mathematical theory of functional dynamics of cortical and thalamic nervous-tissue. Kybernetik 13: 55–59
Wright JJ, Liley DTJ (1996) Dynamics of the brain at global and microscopic scales: Neural networks and the EEG. Behav Brain Sci 19: 285
Xu WF, Huang XY, Takagaki, Wu JY (2007) Compression and reflection of visually evoked cortical waves. Neuron 55: 119
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Robinson, P.A. Neural field theory with variance dynamics. J. Math. Biol. 66, 1475–1497 (2013). https://doi.org/10.1007/s00285-012-0541-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00285-012-0541-x