Dynamical complexity in a mean-field model of ... - Semantic Scholar

Dynamical complexity in a mean-field model of human EEG Federico Frascolia , Mathew P. Dafilisa , Lennart van Veenb , Ingo Bojakc , and David T.J. Lileya a Brain

Sciences Institute (BSI), Swinburne University of Technology, P.O. Box 218, Victoria 3122, Australia; b Department of Mathematics and Statistics, Faculty of Arts and Sciences, Concordia University, 1455 de Maisonneuve Blvd. W., H3G 1M8 Montreal, Quebec, Canada; c Department of Cognitive Neuroscience (126), Donders Institute for Neuroscience, Radboud University Nijmegen Medical Centre, Postbus 9101, 6500 HB Nijmegen, The Netherlands; ABSTRACT

A recently proposed mean-field theory of mammalian cortex rhythmogenesis describes the salient features of electrical activity in the cerebral macrocolumn, with the use of inhibitory and excitatory neuronal populations (Liley et al 2002). This model is capable of producing a range of important human EEG (electroencephalogram) features such as the alpha rhythm, the 40 Hz activity thought to be associated with conscious awareness (Bojak & Liley 2007) and the changes in EEG spectral power associated with general anesthetic effect (Bojak & Liley 2005). From the point of view of nonlinear dynamics, the model entails a vast parameter space within which multistability, pseudoperiodic regimes, various routes to chaos, fat fractals and rich bifurcation scenarios occur for physiologically relevant parameter values (van Veen & Liley 2006). The origin and the character of this complex behaviour, and its relevance for EEG activity will be illustrated. The existence of short-lived unstable brain states will also be discussed in terms of the available theoretical and experimental results. A perspective on future analysis will conclude the presentation. Keywords: electroencephalogram, chaos, Shilnikov saddle node, homoclinic doubling cascade

1. INTRODUCTION The scalp recordable electrical activity (electroencephalogram or EEG) produced by the underlying cerebral cortex is one of the most prominent and robust features of the working brain. Since its first recording from humans by Hans Berger in the late 1920’s1 it has assumed vital phenomenological importance due to its often specific and sensitive correlation with a range of behavioural and pharmacological states. Of particular significance are the waxing and waning 8 - 13 Hz oscillations known as the alpha rhythm which are typically recorded from electrodes placed on the back of the head. The amplitude of these oscillations are typically maximal during a state of wakeful restfulness when the eyes are closed, and are heavily attenuated when the eyes are opened or when some form of mental/cognitive effort occurs. Because the alpha rhythm and other forms of rhythmic EEG activity are recorded from electrodes that sample large territories of underlying cortex, of the order of squares of centimeters, it has been often assumed that the EEG only coarsely reflects brain state and as such its study will not be particularly revealing as to how brains process information and produce behaviour. Implicit in this assumption is that the biological genesis of these oscillations is understood. However despite 80 years of detailed phenomenological study we remain essentially ignorant regarding the mechanisms underlying the genesis of these “bulk” or “molar” oscillations and their relevance to brain information processing and function.2 Because of the staggering diversity of the, often contradictory, empirical phenomena associated with alpha and other forms of EEG activity, the notion of finding simple and consistent biological causes is probably a desideratum not in sight. The complexity of the phenomena to be explained instead necessitates the use of mathematical models and computer simulations, constrained by known physiology and anatomy, in order to understand the richness of the emerging electroencephalographic dynamics. To date models and theories of the EEG can be divided into two complementary kinds. The first assumes that the cortical tissue producing the Further author information: (Send correspondence to D.T.J.L.) D.T.J.L..: E-mail: [email protected], Telephone: 61 3 9214 8812 Biomedical Applications of Micro- and Nanoengineering IV and Complex Systems, edited by Dan V. Nicolau, Guy Metcalfe Proc. of SPIE Vol. 7270, 72700V · © 2008 SPIE · CCC code: 1605-7422/08/$18 · doi: 10.1117/12.813966

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scalp recordable electrical activity is best described by the activity of enumerable, and hence discrete, networks of individual elements (neurons). While these models are useful in the context of sufficient microscopic physiological detail, they are limited since the EEG is a bulk property of populations of cortical neurons.3 A single scalp EEG electrode is sampling the activity of many hundreds of thousands of cortical neurons. Instead a more preferable approach exists which is variously referred to as the continuum or mean field method.4–8 Here it is the bulk or population activity of a region of cortex that is modelled. These models are particularly attractive as their formulation and parametrisation is sufficiently flexible to take into account existing, and probably continuing, uncertainties in cortical anatomy and physiology. The construction of a mean field model of cortical neuronal activity depends upon three essential assumptions: i) the number of neuronal populations modelled. The number of morphologically distinct neuronal populations has, since the pioneering histological studies of Golgi and Ramon y Cajal, been shown to be large. However functionally the number is much less and can essentially be divided into those neurons that excite other neurons (excitatory neurons) and those neurons that inhibit other neurons (inhibitory neurons), ii) the degree of physiological complexity modelled for each neuronal population, and iii) the inter-connectivity of the various neuronal populations. While the majority of mean field theories of EEG model the dynamics of at least two cortical neuronal populations (excitatory and inhibitory), the details of such connectivity can vary substantially, with the consequence that their respective dynamical behaviour can be qualitatively quite different. One mean field model is of particular interest. The model of Liley et al 4, 9, 10 is a relatively comprehensive model of the electrencephalographically dominant alpha rhythm, in that it is capable of reproducing the main spectral features of spontaneous (i.e. not locked to some stimulus) EEG in addition to being able to account for a number of qualitative and quantitative EEG effects induced by a range of pharmacological agents, such as the minor tranquillisers and volatile general anaesthetic agents. However in addition to being of physiological interest this theory is also of significant mathematical and physical interest as it strongly suggests that the human alpha rhythm represents a dynamically critical state from which a range of mathematically unusual bifurcations to complexity (chaos) can occur. One of these bifurcations to chaos was first proposed by Leonid Shilnikov as early as 1969 but has never been found before in a physical theory.11 Here we outline the theoretical basis for this model of electrorhythmogenesis, the qualitative organization of a physiologically admissible parametric subspace which gives rise to electroencephalographically relevant and mathematically interesting complexity, and the possible implications of such parametric structuring for better understanding the dynamics of normal brain function.

2. A MACROSCOPIC MODEL OF EEG AND ITS VARIOUS REDUCTIONS Like many other models, the Liley model considers the dynamics of two interacting spatially distributed cortical neuronal populations. Each of the neuronal populations has the spatial extent of a barrel-shaped region (macrocolumn) of approximately 0.5-3 mm in diameter penetrating the entire thickness of cortex (≈ 3 − 4 mm). The cortical activity is locally described by the mean soma membrane potentials of the excitatory neuron population, he , and the inhibitory neuron population, hi , along with the mean synaptic activities Iee , Iie , Iei , Iii , each modeling the interaction between spatially distributed excitatory (e) and inhibitory (i) neuronal populations. Double subscripts indicate first source and then target. The connection with physiological measurement is through he , which on the basis of extensive experimental evidence, is assumed to be linearly related to the EEG. Individual neurons are modelled as passive RC compartments into which all synaptically induced ionic currents terminate. Thus the respective mean soma membrane potentials are driven from their resting values by eq the synaptic activity weighted by the corresponding ionic driving forces heq jk − hk , where hek,ik are the reversal potentials for excitatory and inhibitory synaptic activity in neurons of type k = e, i. The equations describing the dynamics of he and hi are ∂ he ∂t ∂ τi h i ∂t

τe

heq heq ee − he ie − he I I , + ee eq r ie |hee − hre | |heq ie − he | heq − he heq ii − he I = hri − hi + ei + ei eq r Iii . |hei − hri | |heq ii − hi | = hre − he +


(1) (2)

where hre and hri are the mean resting potentials for excitatory and inhibitory neurons respectively, and τe and τi are the respective neuronal population membrane time constants. Note that the driving forces are normed to one at hj = hrj . The mean synaptic activities, Iee , Iie , Iei , Iii , describe the postsynaptic activation of ionotropic neurotransmitter receptors by presynaptic action potentials arising from the respective inhibitory and excitatory neuronal populations. The time course of such activity, based on extensive experimental evidence, is modelled by a critically damped oscillator driven by the mean rate of incoming excitatory or inhibitory axonal pulses. Thus for excitatory and inhibitory neuronal activity we have for k = e, i

2 ∂ + γek Iek ∂t 2 ∂ + γik Iik ∂t

=

β exp(1)γek Γek Nek Se + pek + Φek ,

(3)

=

β exp(1)γik Γik Nik Si + pik .

(4)

where γj are the respective rate constants that characterise the time course of synaptic activity. For a single presynaptic pulse, modelled by a Dirac delta function, the modelled synaptic time course corresponds to the unitary postsynaptic potential (PSP), with 1/γjk and Γjk being the corresponding time-to-peak and amplitude at peak respectively. The terms in square brackets on the RHS represent the different classes of sources of the presynaptic action potentials: Sj represent action potentials originating locally (i.e within the same macrocolumn), pjk represent extracortical (principally thalamic) sources and Φek the excitatory action potential arriving via the long range cortico-cortical fibres that originate in other macrocolumns. Because only excitatory neurons project over long distances there are no Φik . If time delays for the propagation of action potential within a macrocolumn are considered negligible, then the mean rate of incoming action potentials β times the current local mean neuronal from local sources will be the number of local synaptic connections Njk firing rate Sk , which is assumed to be a sigmoidal function of the respective mean soma membrane potential i.e. Sk ≡ Sk (hk ): √ hk − µk −1 Sk = Skmax 1 + exp − 2 . σk

(5)

where Skmax is the maximum mean neuronal firing rate and µk and σk describe the location and slope of the point of inflection of the firing rate function. Because action potentials arising via the cortico-cortical fibre system traverse much larger distances than the local (intra-columnar) axonal fibres, their conduction delay can no longer be considered negligible and thus they will propagate with finite conduction velocity. This, together with the well established anatomical principle that cortical areas further away will share fewer long ranger connections, means that we are able to describe the propagation of excitatory axonal activity to good approximation via the following inhomogeneous, two-dimensional telegraph equation:

1 ∂2 2Λek ∂ 2 2 α 2 − ∇ + Λek Φek = Nek Λek Se , 2 ∂t2 + v vek ek ∂t

(6)

α where Λek is the characteristic length of the cortico-cortical fibres, vek is their conduction velocity and Nek are the number of long range connections per excitatory cortical neuron.

Equations (1)–(6) are the constitutive equations for the Liley model. An important feature of this model is that there are no ’toy parameters’ in the constitutive equations, i.e. every parameter has a biological meaning and its range can be constrained by physiological and anatomical data. Within the ‘physiological’ ranges of the parameters all sorts of model dynamics can be encountered. However, in order to assess the value of the theory in accounting for essential features of the human electroencephalogram, such as alpha activity, physiologically admissible parametrisation need to produce electroencephalographically plausible dynamics. In general two approaches can be conceived of to generate such parameter sets and to evaluate


them for physiological plausibility. The first is to attempt to fit the model to real electroencephalographic data. While attempts have been made to fit (or estimate) sets of ordinary differential equations to (from) experimentally obtained data, there is still considerable uncertainty regarding the reliability, applicability and significance of these approaches and their relevance to modeling EEG dynamics.12 Alternatively one can develop various approaches to exploring the physiologically admissible multi-dimensional parameter space in order to identify parameter sets that give rise to ‘suitable’ dynamics e.g. those showing a dominant alpha rhythm. In regards to the extended Liley model outlined in the previous section three broad methods declare themselves. Firstly, one could stochastically or heuristically explore the parameter space of the full set of defining partial differential equations using numerical simulations. However, even for multi-parallel simulations, this is too slow for either iterative parameter optimization or Monte-Carlo search. Secondly, the parameter space of the spatially homogeneous cortical model obtained by equating the Laplacian in equation 6 to zero can be numerically searched. Finally, if the defining system can be approximated well by linearization for moderate noise input, then one can estimate all relevant dynamics, and their spatial dependency, merely from the corresponding eigenvalues and eigenvectors. Such an eigen-analysis is exceedingly rapid compared with the direct solution of the equations. One can then abandon any form of iterative constrained optimization and simply test parameter sets randomly sampled from the physiologically admissible parameter space. Thus, for example Bojak and Liley (2005)13 show how one can model plausible EEG recorded from a single electrode using this approach. The obvious drawback of this approach is that non-linear solutions of potential physiological relevance will not be obtained. However, as will be illustrated in the next section, this surprisingly is not the case, as one and two-dimensional dynamical continuations of parameter sets obtained on the basis of their eigenspectra, reveal a plethora of electroencephalographically plausible non-linear dynamical behaviour.

3. MODEL DYNAMICS, BIFURCATIONS AND ROUTES TO COMPLEXITY Numerical solutions to equations (1)–(6) for a range of physiologically admissible parameter values reveal a large array of deterministic and noise driven dynamics, and bifurcations, especially at alpha band frequencies.4, 9, 10, 14 In this regard, alpha band activity appears in three distinct dynamical scenarios: as linear noise driven, limit cycle or chaotic oscillations. Thus this model offers the possibility of characterizing and explaining not only the genesis of alpha band activity, but also the changes in its qualitative dynamics that have been inferred to occur during cognition12 and in a range of central nervous system disease processes such as epilepsy.15 Further, our theory predicts that reverberant activity between inhibitory neuronal populations is the dynamical source of the alpha rhythm and as a consequence the strength and form of population inhibitory→ inhibitory synaptic interactions will be the most sensitive determinants of the frequency and damping of the emergent alpha band oscillatory activity. In particular, if resting eyes-closed alpha is indistinguishable from a filtered random linear process, as some time series analyses seem to suggest (see Stam (2005)16 for a review), then our model implies that electroencephalographically plausible “high Q” alpha (FWHM > 5) can only be obtained in a lumped system in which there exists a conjugate pair of weakly damped (marginally stable) poles at the alpha frequency.4 Numerical analysis has revealed regions of parameter space where abrupt changes in alpha dynamics occur. Mathematically these abrupt changes correspond to bifurcations, whereas physically they might be conceived as resembling phase transition phenomena in ordinary matter. Figure 1 reveals such a region of parameter space for the 10-dimensional homogeneous reduction of equations (1)–(6) (equations 1–5 with Φek = 0). Variations in pee (excitatory input to excitatory neurons) and pei (excitatory input to inhibitory neurons) are seen to result in the system producing a range of dynamically differentiated alpha activity. First of all, if pei is much larger than pee a stable equilibrium is the unique state of the EEG model: driving the model in this state with white noise typically produces sharp alpha resonances.4 If we increase pee , this equilibrium looses stability in a Hopf bifurcation and periodic motion sets in with a frequency in the alpha range of about 11 Hz. For larger pee the fluctuations can become irregular and the limiting behaviour of the model is governed by a chaotic attractor. The different dynamical states can be distinguished by computing the largest Lyapunov exponent (LLE), which is negative, zero, or positive for equilibria, (quasi)-periodic fluctuations, and chaotic fluctuations, respectively. Bifurcation analysis11 indicates that the boundary of the chaotic parameter set is formed by infinitely many saddle-node and period-doubling bifurcations as shown in Figure 1(a): all these bifurcations converge to a narrow wedge for


(a)

(b) region in which chaotic activity terminates

Hopf

chaos

gh

SN (eq.)

c

SN (per. orbit ) period doubling homoclinic

n1 n2

bt

t1

(a)

(c)

t2

n1 t2 n2 t1 maximum of h e

Figure 1. (a) The largest Lyapunov exponent (LLE) of the dynamics of a simplified version of equations (1)–(6)(equations 1–5 with Φek = 0) for a physiologically plausible parameter set exhibiting robust (fat-fractal) chaos.10 Superimposed is a two parameter continuation of saddle-node and period-doubling bifurcations. The leftmost wedge of chaos terminates for negative values of the exterior forcings, pee and pei . (b) Schematic bifurcation diagram at the tip of the chaotic wedge. BT = Bogdanov-Takens bifurcation, GH = generalized Hopf bifurcation and SN = saddle node. Between t1 and t2 multiple homoclinic orbits coexist and Shilnikov’s saddle node bifurcation takes place. (c) Schematic illustration of the continuation of the homoclinic orbit between points n1 and t1 . Between t1 and t2 multiple homoclinic orbits exist and Shilnikov’s saddle node bifurcation takes place. Figure adapted from van Veen and Liley (2006)11 and Dafilis et al (2001).10


negative, and hence un-physiological, values of pee and pei , literally pointing to the crucial part of the diagram where a Shilnikov saddle-node homoclinic bifurcation takes place. Figure 1(b) shows a schematic blowup of the bifurcation diagram at the tip of the wedge: the blue line with the cusp point c separates regions with one and three equilibria, and the line of Hopf bifurcations terminates on this line at the Bogdanov-Takens point bt. The point gh is a generalised Hopf point, where the Hopf bifurcation changes from sub- to super-critical. The green line which emanates from the bt represents a homoclinic bifurcation, which coincides with the blue line of saddle-node bifurcations on an open interval, where it denotes an orbit homoclinic to a saddle node. In the normal form, this interval is bounded by the points n1 and n2 , at which points the homoclinic orbit does not lie in the local center manifold. While the normal form is twodimensional and only allows for a single orbit homoclinic to the saddle-node equilibrium, the high dimension of the macrocolumnar EEG model (equations 1–5 with Φek = 0) allows for the several orbits homoclinic to the saddle node. If we consider the numerical continuation of the homoclinic along the saddle node curve, starting from n1 , as shown in Figure 1(b), it actually overshoots n2 and folds back at t1 , where the center-stable and center-unstable manifolds of the saddle node have a tangency. In fact, the curve of homoclinic orbits folds several times before it terminates at n2 , thereby creating an interval, bounded by t1 and t2 , in which up to four homoclinic orbits coexist, signalling the existence of infinitely many periodic orbits: this is the hallmark of chaos. It is important to appreciate that, in contrast to the homoclinic bifurcation of a saddle focus, commonly referred to as the Shilnikov bifurcation, this route to chaos has not been reported before in the analysis of any mathematical model of a physical system. While the Shilnikov saddle node bifurcation occurs at negative, and thus unphysiological, values of pee and pei , it nevertheless organizes the qualitative behaviour of the EEG model in the biologically meaningful parameter space. Further, let us remark that this type of organization persists in a large part of the parameter space: if a third parameter is varied, the codimension-two points c, BT and GH collapse onto a degenerate Bogdanov-Takens point of codimension three, which represents an organizing center controlling the qualitative dynamics of an even larger part of the parameter space. Importantly, parameter sets that have been chosen to give rise to physiologically realistic behaviour in one domain, can produce a range of unexpected, but physiologically plausible, dynamical activity in another. For example it has been found that parameter sets chosen to accurately model eyes-closed alpha and the surge in total EEG power seen during anaesthetic induction can, under appropriate parameter perturbations, also give rise to gamma band (> 30 Hz) oscillations, activity thought to be the conditio sine qua non of cognitive functioning.14 Specifically parameter sets were chosen on the basis of the physiological plausibility of their eigenspectra, which required among other things having a sharp alpha resonance (FWHM > 5) and mean excitatory and inhibitory neuronal firing rates < 20 s−1 (see Bojak and Liley (2005)13 for full details). Surprisingly, a large fraction of sets were also able to produce limit cycle (non-linear) gamma band activity under relatively small perturbation in a parametric subspace. Theoretically at least, this suggests that the existence of weakly damped, noise driven, linear alpha activity can be associated with limit cycle 40 Hz activity and that transitions between these two dynamical states can occur. Figure 2(a) illustrates a bifurcation diagram for one such set (column 11 of Table V in Bojak and Liley (2005);13 also see Table 1 in Bojak and Liley (2007)14 ) for equations (1)–(6) with ∇2 → 0. The choice of bifurcation parameters is in this case motivated by two observations - (i) differential increases in Γii,ie have been shown to reproduce a shift in alpha to higher beta band EEG activity that is seen in the presence of low levels of GABAA agonists such as the benzodiazepines17 and (ii) the dynamics of linearized solutions to equations (1)–(6) are particularly sensitive to variations in the parameters affecting inhibitory→inhibitory neurotransmission4 such as Niiβ and pii . Specifically, Figure 2 illustrates the results of a two parameter bifurcation analysis for changes in the inhibitory PSP amplitudes via Γie,ii → rΓie,ii and changes in the total number of inhibitory→inhibitory connections via Niiβ → kNiiβ . The parameter space assumes a physiological meaning only for positive values of r and k. The saddle-node bifurcations of equilibria have the same structure as for the 10-dimensional homogeneous reduction discussed previously, in that there are two branches joined at a cusp point. Furthermore, we have two branches of Hopf bifurcations, the one at the top being associated with the birth of alpha frequency limit cycles and the other with gamma frequency limit cycles. This former line of Hopf points enters the wedge shaped curve of saddle-nodes of equilibria close to the cusp point and has two successive tangencies in fold-Hopf points (fh), which are connected by a line of tori or Neimark-Sacker bifurcations of periodic cycles. The same curve of


(a) 1.5

saddle node (equilibrium) Hopf torus saddle node (cycle) period doubling homoclinic

fh

k

1

gh

fh 0.5

bt

0

0.02

cpo

0.06

0.1

0.2

0.4

0.6

0.8 1

R

(b)

Figure 2. (a) Partial bifurcation diagram for the spatially homogeneous PDE model (equations 1–6 with ∇2 → 0) as a function of the two scaling parameters k and r, defined respectively by Γie,ii → rΓie,ii and Niiβ → kNiiβ . The codimension two points have been labeled fh for fold-Hopf, gh for generalized Hopf (transition form sub- to super-critical Hopf bifurcation) and bt for Bogdanov-Takens. The right-most branch of Hopf points corresponds to the emergence of gamma frequency (≈ 37Hz) limit cycle activity via a subcritical Hopf bifurcation above the point labelled gh. A homoclinic doubling cascade takes place along the line of homoclinics emanating from bt. (b) Numerical solutions of the full set of equations (equations 1–6) for a human sized toroidal cortical sheet for r = 0.875. The mean excitatory soma membrane potential, he , is mapped every 60 ms (grayscale: -76.9 mV, black to -21.2 mV, white). For r = 0.875 the linearization of equations (1)–(6) becomes unstable for a range of wavenumbers around 0.6325cm−1 . From random initial conditions in he , synchronised gamma activity emerges and spatial patterns with a high degree of phase correlation (gamma ‘hotspots’) form having a frequency consistent with the predicted subcritical Hopf bifurcations seen in the spatially homogeneous equations (see (a)). Figure (b) reproduced from Bojak and Liley (2007).14


Figure 3. One of the chaotic attractors whose presence is due to the appearance of a homoclinic doubling cascade in the model. Top figure: the chaotic time series drawn in the he -hi plane, corresponding to the parameters (r, k) ≈ (0.293, 0.537). Bottom figure: the power spectrum associated with the time series: a peak in the alpha range around 11Hz and subharmonics are clearly evident.

Hopf points ends in a Bogdanov-Takens (bt) point, from which a line of homoclinics emanates. Contrary to the previous example, the interaction between homoclinics and period doublings bifurcations gives rise to a so-called homoclinic doubling cascade, where infinitely many periodic orbits and homoclinics connections are created.18 This mechanism leads to the appearance of chaotic attractors, an example of which is depicted in Figure 3. Finally, the second line of Hopf bifurcations in the gamma frequency range (> 30Hz) does not interact with the lines of saddle nodes in the relevant portion of the parameter space. Both branches of Hopf points change from sub- to supercritical at gh around r∗ = 0.27, so that bifurcations are “hard” for r > r∗ in either case. These points are also the end points of folds for the periodic orbits, and the gamma frequency ones form a cusp (cpo) inside the wedge of saddle nodes of equilibria.

4. CONCLUSION - NEW MATHEMATICS AND PHYSIOLOGICAL INSIGHT The occurrence of a Shilnikov saddle-node bifurcation in a mathematical model which plausibly describes the main features of EEG dynamics is undoubtedly a relevant result from the mathematical point of view. At the same time, the analysis of local models points to the existence of similar invariant dynamical structures in the full partial differential equation mode. This means that, as a number of generic and thus detectable bifurcations are organized about the Shilnikov saddle-node, an experimental strategy can be suggested by which the parametric organization of these bifurcations, for example in cortical slice preparations, can be used to experimentally infer whether real mammalian cortex is capable of supporting chaos. This may provide an alternative to the largely unsuccessful attempts19 to identify chaotic activity directly from noisy and nonstationary time series. The challenge for experimentalists will thus be to systematically manipulate pee and pei to assume both positive and negative (unphysiological) values, and this might be achieved using M-type current (a slowly activating persistent inward ionic membrane current), antagonists (e.g., carbachol) and agonists (e.g., retigabine). When cortico-cortical fibres are considered, the general mechanism of presynaptic inhibition or presynaptic facilitation is a strong candidate for the explanation of the complex nonlinear behaviour associated with the homoclinic doubling cascade. As the fraction of inhibitory synapses in cortex is about the same order as


the number of thalamic terminals,20 it is possible that these might reduce effective inhibitory PSP amplitudes through the inhibition of presynaptic GABA release, which has been modelled by reducing Γie,ii . Alternatively, excitatory thalamic afferents may excite a range of layer IV interneurons, which presynaptically facilitate inhibitory-inhibitory connectivity, enhancing the presynaptic release of GABA, modelled by raising Niiβ . This second mechanism seems more plausible as hippocampal interneurons have been shown to enhance the presynaptic release of GABA.21 Correlated gamma band activity indicative of large scale integration can emerge spontaneously for both cases in our physiological theory of EEG, and the interplay between the two is responsible for the rich bifurcation picture of the underlying dynamics. New work is underway in mainly three directions. First, other parameter sets within physiologically relevant ranges, that are likely to show routes to chaos alternative to the ones described in this paper, are currently under investigation. Secondly, we are interested in assessing the role of cortico-cortical connections in a systematic way, in terms of the dynamical states that the model is allowed to display, especially in regards to the 40 Hz gamma buzz associated with cognition. Finally, the bifurcation analysis for the full PDE version of the Liley equations will take spatial inhomogeneities of the solutions into account and possibly shed new light on the genesis and functional significance of the complexity that lies at the heart of brain dynamics.

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