Quantitative theory of driven nonlinear brain dynamics J. A. Robertsa,b,c,⁎, P. A. Robinsona,b a School of Physics, University of Sydney, New South Wales 2006, Australia b Brain Dynamics Center, Sydney Medical School —Western, University of Sydney, Westmead, New South Wales 2145, Australia c Queensland Institute of Medical Research, Brisbane, Queensland 4006, Australia abstract article info Article history: Accepted 21 May 2012Available online 29 May 2012 Keywords:Steady state visual evoked potential Entrainment Neural ﬁeld model Nonlinear dynamics EEGPeriodic stimulus Strong periodic stimuli such as bright ﬂashing lights evoke nonlinear responses in the brain and interact nonlinearly with ongoing cortical activity, but the underlying mechanisms for these phenomena are poorly understood at present. The dominant features of these experimentally observed dynamics are reproducedby the dynamics of a quantitative neural ﬁeld model subject to periodic drive. Model power spectra over a range of drive frequencies show agreement with multiple features of experimental measurements, exhibiting nonlinear effects including entrainment over a range of frequencies around the natural alpha frequency f α, subharmonic entrainment near 2 fα, and harmonic generation. Further analysis of the driven dynamics as a function of the drive parameters reveals rich nonlinear dynamics that is predicted to be observable in future experiments at high drive amplitude, including period doubling, bistable phase-locking, hysteresis, wave mixing, and chaos indicated by positive Lyapunov exponents. Moreover, photosensitive seizures arepredicted for physiologically realistic model parameters yielding bistability between healthy and seizure dy-namics. These results demonstrate the applicability of neural ﬁeld models to the new regime of periodically driven nonlinear dynamics, enabling interpretation of experimental data in terms of speci ﬁc generating mechanisms and providing new tests of the theory. © 2012 Elsevier Inc. All rights reserved. Introduction Periodic ﬂashing light, also termed ﬂicker or photic driving, evokes responses in the electroencephalogram (EEG) that have been studied since the 1930s. Despite being widely used to probe visual function and perception ( Müller et al., 2006; Nunez, 1995; Regan, 1989 ), the un-derlying neurophysiological mechanisms of these responses remain poorly understood. Early studies identi ﬁed a clear nonlinear contribu-tion to the driven cortical dynamics ( van der Tweel and Spekreijse, 1969; van der Tweel and Verduyn Lunel, 1965 ). Nonlinearity in large-scale brain activity has seen considerable recent interest ( Stam, 2005 ), particularly in healthy resting state activity ( Freyer et al., 2009, 2011; Stam et al., 1999 ), epileptic seizures ( Breakspear et al., 2006; Lehnertz and Elger, 1998; Robinson et al., 2002; Wendling et al., 2000 ), and func-tional neuroimaging ( Stephan et al., 2008a ). Notable nonlinear features in the EEG response to ﬂicker include entrainment and frequency mixing ( Herrmann, 2001; Rager and Singer, 1998; Regan, 1989; Townsend et al., 1975 ), both of which involve interactions between stimuli and ongoing activity and are thus of central importance to un-derstanding brain function ( Engel et al., 2001 ). Periodic stimuli can also provoke epileptic seizures ( Parra et al., 2005 ), particularly for driving frequencies near the natural alpha rhythm fα(≈10 Hz in adult humans), the most prominent feature of healthy resting EEG ( Nunez, 1995 ). The mechanisms of these seizures are not well understood, nor is there a clear relation to the brain's re-sponse in cases when the stimuli do not cause seizure. Thus there is a need for modeling to unify these nonlinear phenomena within a sin-gle framework. Studies of the nonlinear mechanisms responsible for ﬂicker-induced EEG signals have primarily taken a signal analysis approach ( Kelly, 1981; Liu et al., 2010; Regan, 1989; van der Tweel and Spekreijse, 1969; van der Tweel and Verduyn Lunel, 1965 ), modeling time series data as arising from abstract ﬁlters. Although successful in describing input-output relationships, such models are dif ﬁcult to generalize to other phenomena; it is preferable to have a single systematic theory. In-deed there is a strong need in neuroscience to move beyond models of time series to models of the underlying brain structures and connectiv-ity, from which the observed dynamics emerge ( Breakspear et al., 2010; Stephan et al., 2008b ). Such modeling is now possible given recent de-velopments in neural ﬁeld theory ( Coombes, 2010; Deco et al., 2008 ), particularly as it relates to the corticothalamic system ( Robinson et al., 1997, 2002, 2004 ). Neural ﬁeld models describe the aggregate activity of populations of neurons and are thus especially appro priate for describing large-scale brain dynamics. Building on early work ( Amari, 1975; Ermentrout and Cowan, 1979; Freeman, 1975; Lopes da Silva et al., 1974; Nunez, 1974;Neuro Image 62 (2012) 1947-1955 ⁎Corresponding author at: Queensland Institute of Medical Research, 300 Herston Road, Herston, Queensland 4006, Australia. Fax: +61 7 3845 3511. E-mail addresses: jamesr@physics. usyd. edu. au (J. A. Roberts), robinson@physics. usyd. edu. au (P. A. Robinson). 1053-8119/$-see front matter © 2012 Elsevier Inc. All rights reserved. doi:10. 1016/j. neuroimage. 2012. 05. 054 Contents lists available at Sci Verse Science Direct Neuro Image journal homepage: www. elsevier. com/locate/ynimg