The Impact of Chaos on Science and Society (UNU, 1997, 394 p.) |
Arnold J. Mandell and Karen A. Selz
I. Introduction
At Professor Hermann Haken's 1982 synergetics meeting in Schloss Elmau, the first author suggested to Doyne Farmer, Gottfried Mayer-Kress, and Agnes Babloyantz that the electroencephalographic record of cerebral electrovoltage fluctuations (EEG) taken from the brains of animals and man appeared to be a "strange attractor" (see Mayer-Kress 1986, for an account of this conference). Our laboratory had begun phase space reconstructions and measures of complexity on time series of a variety of neurophysiological observables and compared our findings with conventional frequency (power) spectra (Mandell et al. 1982). The "similarity" in the changes in these measures invoked by pharmacological perturbation across neural systems operating at a wide range of time-scales suggested that the natural fractal structure of strange attractors might be the dynamical mechanism responsible for the integration of brain mechanisms across nested hierarchies of characteristic times.
Did the structure of chaos explain how it was that neurotransmitter enzyme mechanisms in microseconds, neural mechanisms in mille-seconds, synaptic mechanisms in seconds, adaptive regulation in minutes, and behavioural changes in hours, days, weeks, and months might be coordinated (Mandell et al. 1981,1983)? The spawning of a global hierarchical horseshoe cascade by homoclinic intersection is a candidate for a deterministic global dynamic that could generate "zoom" invariant, self-affine fractal orbits that one could imagine represented scaling levels of the brain's temporal hierarchy. It is also clear that a variety of forms of the generalized Hodgkin-Huxley equations for membranes and neurons manifested parameter spaces with both intermittent and chaotic dynamics (Carpenter 1981; Rinzel 1987; Aihara and Matsumoto 1987; Selz and Mandell 1991).
The EEG "diagnosis" of a strange attractor was consistent with another hypothesis that we were pursuing at the time: why was it that a number of disease states in biological dynamical systems manifested themselves via a loss in complexity (Mandell 1981; Goldberger et al. 1986; Mandell 1987)? It seemed natural that a deterministic, information-generating mathematical object such as a strange attractor would be consistent with the healthy EEG and the use of this positive entropy supply by the brain for information-bearing pattern formation.
Since that time it has been shown by Babloyantz and her colleagues (1986, 1988) that (petit mal) epilepsy and a case of hereditary cerebral degeneration with dementia were associated with the expected decrease in complexity of the EEG signal. We treat their "correlation dimension," D_{2}, in the modern Renyi parlance (1959), as a third moment of the entropy (after D_{0} and D_{1}) without defining it further at this time. Other studies using this measure on the EEG seemed to support the notion that a complexity decrement is associated with decreases in alertness and intellectual performance by the brain. For example, the deeper the sleep, the lower the measure of complexity of the EEG (Babloyantz et al. 1985; Roschke and Basar 1989; Roschke and Aldenhoff 1991). At the other end of central nervous system function are the findings of Rapp et al. (1989), in which the D_{2} of the EEG is transiently increased in some subjects performing cognitive tasks that require recognition and response to novelty.
There is a range of D_{2} values for the EEG that are generally lower if the singular value decomposition technique is applied (to obtain the major eigenvectors). For example, in states of "relaxed, awake," it generally varies from fractional exponents with values of 3 to 6+. It is clear that these numbers may mean little with respect to absolute value but serve as an index of "complexity" in test-treatment-retest experimental designs. A recent example of a comparative application is the finding of an overall D_{2} value of a little over 4 that increased in the transition from eyes closed to eyes open but less so in the elderly and much less so in the elderly with Alzheimer's dementia (Prichard et al. 1991). It is possible that some absolute number has specific meaning. We will suggest below that it may be that the dimensionality is an estimate of the value for n in the space of EEG dynamics, T^{n} and that a strange attractor theory of the global EEG signal is not a good guess.
These empirical successes with respect to the prediction of brain states from the EEG using the now standard embedding-correlation dimension algorithm grew out of the hyperbolic strange attractor theory of the Smale school (1967), the entropic isomorphism ideas of Ornstein (1974), and the fractal (Parieto-Levy) statistical measure theory of Mandelbrot (1977). It behoves us to ask whether these new techniques add to our understanding of the EEG. It is when we begin to move deeper into the implications of a strange attractor theory of the EEG for neurobiological mechanisms, in relationship to what is known about the EEG, that another theory of the dynamics of the EEG emerges, n-quasiperiodic modes on T^{n}, which may, by current technology, be impossible to disentangle from the strange attractor theory.
The arguments concerning the appropriate theoretical model for the EEG centre around those involved with the original theoretical break from the Landau-Hopf theory of hydrodynamic turbulence by the work of Ruelle and Takens (1971) and Newhouse, Ruelle, and Takens (1978). The Landau-Hopf picture is that of increasing Reynold stress generating a countably infinite number of incommensurable frequencies on T^{n}, n ® ¥, at the onset of turbulence. The Newhouse-refined Ruelle-Takens picture on n ³ 4, suggested that the stability issues of genericity were such that chaotic solutions were dense in the neighbourhood of all quasiperiodic solutions for n ³ 3 and that they (in contrast to quasiperiodic orbits easily perturbed to mode locking on one hand or chaotic solutions on the other) were structurally stable.
The issues of genericity and structural stability in n-tori has long been a theoretical problem of mathematical depth (see below) but it is of interest that an "abundance" of 3-tori have been found in both experimental turbulence (Gollub and Benson 1980) and numerical simulations using two coupled, periodically driven Van der Pol systems (Battelino et al. 1989). There are three spatial-functional symmetrics in the brain: anterior-posterior (of future and past, of imagining versus seeing), right-left, (analytic-algebraic versus geometric-intuitive) and up and down the neural axis (more or less complexity), which oscillating in time with characteristic modes make issues of expected versus exceptional bifurcation behaviour different than they would be if studied only in time.
Spatial symmetries of the brain make the "nongeneric" stability of T^{n} in the brain with n ³ 3 more likely (Golibitsky et al. 1988). Measuring zero parameter space phenomena under symmetry conditions can develop positive measure (Knobloch 1990). It will be argued that in the same way that mode locking on T^{2} is prevented by S(02) (circular symmetry of the 2-torus via transformational prevention of the hyperbolicity required by stable mode locking) (Rand 1982), the brain with at least 0(3) symmetry with two independent rotations prevents mode locking and allows T^{4} quasiperiodic mode multiplicity in time. J. D. Crawford (Personal communication) has conjectured that generally, 0(n) symmetry prevents mode locking in T^{n+1} tori via n - 1 independent symmetric rotational transformations. There are many "nongeneric" bifurcation phenomena in fluid mechanics, which have been explained in this way (Crawford and Knobloch 1991) beginning with Ruelle's (1973) argument that S0(2) symmetry in a Hopf bifurcating system generates the rotating waves of Taylor-Couette flow. Analogous to the way that the number of isolated integrals of a Hamiltonian system determines its dimension, the "isolating" symmetries of a dissipative dynamical system may determine the number, n, of the independent frequencies that can be observed in a quasiperiodic system, T^{n}.
We are motivated to seek an alternative to the strange attractor theory of the EEG by such findings as seen in figure 1 (Nunez and Katznelson 1981), which demonstrates how changing concentrations of halothane almost smoothly move the power spectral frequency location of the dominant a-mode of the EEG. This is more reminiscent of the smooth dependence of the rotation number R (f) on f (a homeomorphic reduction of T^{2} to f: S^{1} ® S^{1}) (Herman 1979) than a parametric alternation of a strange attractor. In the circular Couette system, w_{1}/w_{2} is a smooth function of the Reynold's number, this continuous modulation of the frequency content of quasiperiodicity made possible by the circular symmetry of the boundary conditions on T^{2} (German and Swinney 1982). We will argue that the stability of T^{4} in the EEG is made possible by the 0(3) symmetry of the brain.
Fig. 1 Changing concentrations of
halothane move the power spectral frequency location of the dominant a mode of
the EEG
II. The EEG as a global nonlinear oscillator: Quasiperiodic, f-scaling, power-spectral broad band modes
One of the persistent mysteries of the EEG is the source of the slow (by physical standards) rhythms, ranging from 1 to 50+ Hz. The most characteristic frequency bands and their associated states of consciousness are: D, < 4Hz, deep sleep or coma; q, 4-7 Hz, day dreaming and/or light sleep, more prominent in children and young adults with behaviour disorders; a, 10-12 Hz, relaxed, eyes closed, awake, a is "blocked" by increases in attention; b, 18-30+, very alert, aroused, or anxious; g, 35-50 Hz, called "40 Hz," periods, which are transient and thought to be associated with instances of spatial coherence in perceptual tasks. These rhythms are less obvious in the raw record than they are in Fast Fourier Transform of the autocorrelation function, called the frequency (power) spectrum, where these broad band modes are most clearly in evidence. In the context of descriptions of normal EEGs (with the assumed nonlinear coupling in the underlying dynamics) it should be noted that a normal variant is called "the flat EEG" (Pine and Pine 1953), which when magnified to permit its very low amplitude behaviour to be analysed revealed unusually fast frequencies. This is consistent with the amplitude-frequency coupling signature of a nonlinear system.
These facts from over 60 years of research on the human EEG have never been disputed, yet they have never been well understood. Strange attractor theory (see for example Eckmann and Ruelle's discussion of Haken's use of normal mode analysis in complex systems, 1985) has led to serious questions concerning the usual analytic approach to spectra with multiple broad band modes. This involves the decomposition using Fourier series into characteristic frequencies with the ansatz of linear, additive superimposition. In the Haken context, this would amount to the computation of the eigenfunctions (two-dimensional Bessel function or perhaps spherical harmonics) for the "normal modes." The strange attractor theorist views these hierarchical, multi-broad band mode power spectra in another way.
These broad band modes (found in the frequency (power) spectra of the Rossler, Henon, and Lorenz equations, strange or similar-to-strange attractors) are viewed by the "chaos school" as representing the unstable fixed points and/or aperiodic orbits of expanding and folding dynamical systems. They are the result of post-Hopf, global bifurcation phenomena (the scenario of Ruelle and Takens) and are undecomposable. Ruelle (1989) refers to these complex singularities of the power spectrum as "resonances" of the dynamical system and suggests their relation to the decay properties of the time correlation functions, that is, their "mixing" properties.
Although not inconsistent with a strange attractor hypothesis (here this generalized attractor is defined by the presence of unstable fixed points, separation of nearby initial conditions, and phase space inde-composability), it is nevertheless clear that with the use of global invariant measures, such as the Lyapunov characteristic exponent or the Renyi dimension entropies, this characteristic mode structure is lost. The use of a measure like D_{2} adds little to what is already known about systematically varying EEG states with characteristic broad band mode dominance. Is the D_{2} calculation more than another way to compute the average frequency content? For example, with respect to frequency bands of sleep stage (staging exploits the EEG record only as part of the criteria):
D_{2} |
EEG mode |
6.51 ± 0.43 |
Descending stage one 12-14 Hz |
5.67 ± 0.45 |
Stage two 8-10 Hz |
4.69 ± 0.41 |
Stage three 4-6 Hz |
4.25 ± 0.23 |
Stage four 1-3 Hz |
(after Roschke and Aldenhoff 1991). As one might anticipate, the lower the frequency of the dominant broad band mode, the lower the correlation dimension (complexity). The "dimension" D_{2} varies inversely with the statistical "smoothness" of the curve. More formally, a change analogous to one in the Lipschitz constant, k, defined on x, y space (with respect to x) as || f(t,x) - f(t,y)|| £ k ||x - y|| remembering that if the derivative of f for fixed t º D_{x}f(t,x) and ||D_{x}f(t, x)y|| £ k||y|| for all y e R^{Î} then f is Lipschitz with constant k. This property is also referred to as Hr continuity. Increasing differentiability of the EEG planar curve is associated with a decrease in D_{2}. It is also of potential interest that increasing differentiability of this signal is associated with decreased capacity for complexity and variety in brain function.
If one views the D, q, a, b, and g modes of the EEG as approximate, and notes that D + q @ a, q + a @ b and a + b @ g ..., we could analogize this additive proportional-scaling series to that of the Fibonacci numbers. The ratio of mode one, w_{1} to mode two, w_{2}, that is w_{2}/w_{1} @ F = 1.618... (F-^{1} = f = 0.618...). The implication of this proportional relation for a topological dynamical mechanism of the EEG is that this system may be a hierarchy of quasiperiodic modes of the n-torus that is most stable against perturbation-induced (frequency-resonant) mode locking (Arnold 1983). We remember that it is the mode-locked, fixed-point or periodic states that are associated with neurological diseases such as epilepsy and Parkinson's disease (Selz and Mandell 1991). There are two immediately relevant organismic biological aspects of this F-proportionality: (1) How can proportional scaling result in the singular value F? (2) What implication does F-scaling have for the stability (existence) of hierarchical frequencies of the n-torus called quasiperiodicity? Both questions have rather extensive mathematical backgrounds, which we will review only very briefly.
With the simplest conservation assumption about proportionality such that when dividing a normalized range of x, [0,1] into a smaller, s, and larger, l, piece such that if x e [0,1] = s/l,
_{} |
(1) |
then the contraction operator O(x) finds a fixed point x* of the proportionality relation (Dalzell theorem) such that
_{} |
(2) |
in which x* = f = F^{-}^{1} makes the relation an equality and O(x) = 0.
The claim that the dominant modes of the EEG may be quasiperiodic with incommensurate frequency ratios, w_{2}/w_{1} @ F (Arnold 1962, 1983; Moser 1968), rather than the "noisy periodicities" (Lorenz 1980) of systems with sensitivity to initial conditions, involves stability issues contrasting conditions of structurally unstable dynamics of positive measure with those that persist (are structurally stable) with respect to perturbation. The hyperbolic stability arguments of strange attractor theory are well known, but can the alternative quasiperiodic orbits exist? If T^{n}, n = 2, is studied as a surface of section via an orientation-preserving diffeomorphism of the circle, f: S^{1} ® S^{1}, it is known that f is structurally stable (topological equivalence) if and only if _{} = w_{2}/w_{1} is commensurate (e.g., a mode-locked winding number). On the other hand, it is obvious that rational _{} among the real systems have measure zero making their occurrence unlikely in computationally precise systems. This issue is usually resolved for real systems (subject to the statistical desturbances of reality) in favour of the relative stability of rational _{}.
Denjoy's theorem says that if _{} is irrational then it can be changed to a nearby rational _{} by an arbitrarily small perturbation of f and that every diffeomorphism f with irrational _{} is structurally unstable. We know that fixed point (periodic) solutions are associated with brain disorders such as the periodic spiking EEG record of epilepsy, making these structurally stable solutions inconsistent with healthy brain function. Is the "strange attractor" the only alternative?
The proof that quasiperiodic solutions to "reversible" i.e. non-dissipative equations of the general form
_{} |
(3) |
such that if x(t) is a solution, so is x(-t), exist was reported by Arnold (1962) and Moser (1968) for Hamilton's equations of classical mechanics. Moser's conditions involved some bound on the Fourier coefficients and a high order of differentiability, the C^{r} topology where r ³ 5, whereas Arnold's conditions are more relevant to the EEG quasiperiodic mode problem, which involves the choice of irrational _{} with greatest distance from rationals, that is the furthest from mode locking. Using the Euclidean algorithm it can be shown that the partial quotients of the (unending) continued fraction expansions of irrationals are the lowest integers. Using the continued fraction expansion of _{}, of which 1/1 + s/1 (1/1 + x) can be seen as the beginning, Arnold used a limit argument to choose _{} = F with an expansion of [1, 1, 1,...] and went on to prove that quasi-periodic solutions with rationally independent frequencies can form a set of positive measure. These sets are not open and dense, but are complicated Canter sets. In fact, consistent with the studies of diffeomorphisms by Newhouse (1974), there are an infinite number of values of the amplitude of the forcing frequency for which there are stable periodic solutions (mixed in with parameter values inducing quasiperiodic and chaotic solutions).
Whereas these proofs of quasiperiodicity on T^{n} involve the non-dissipation of Hamiltonian mechanics as a condition, sets of positive measure in forced-dissipative systems such as the Cartwright-Little-wood-Van der Pol system (Mandell et al. 1987) (fig. 2) demonstrate large parameter areas of quasiperiodic behaviour (without colour) along with (hatched) "Arnold's tongues" indicating parameter regions of mode-locking behaviour. These studies were done on an EAI-680 analog computer with oscilloscopic monitoring. Systematic parameter space exploration included digitization of the signal by a dedicated PC and statistical evaluation on a Sun 3-160 work station.
Figure 3 portrays the attractors of this nonautonomous, periodically forced nonlinear differential equation in regions of quasiperiodic solutions. The phase portraits of x'/x (fig. 2) look like quasi-periodic orbits with irrational rotation numbers covering the manifold making orbits dense on the torus. The slowing down around the two saddle sinks makes the orbital cover nonuniform with increased Sinai-Bowen-Ruelle measure (time-dependent occupancy) larger in the two regions where x' = 0.
The morphological discrimination of the frequency (power) spectra of quasiperiodic systems with _{} irrational from those of "aperiodic," chaotic dynamics with complex singularities of the dynamics in real systems is far from trivial. If the signal is the sum of independent periodic terms, _{}, then even in a noisy system one can find the major w_{i} and their harmonics. More likely, the signal, x(t) is non-linearly dependent on the periodic functions of the variables w_{i}t and if these are irrationally related, as we have conjectured is the case in the EEG, they will form a dense set over the positive reals. This does not mean that the spectrum is necessarily smoothly continuous, though we will see below it can be, but may be characterized by broad bands and d-functions of a small (finite) number which may reflect the underlying dominant frequency content. Just as likely, if the system involves both the products and sums of circular functions, the mode content will not be so easily understood in terms of the underlying dynamics.
Fig. 2 Sets of positive measure in
forced-dissipative systems demonstrate large parameter areas of quasiperiodic
behaviour in the "Arnold's tongues" (hatched regions)
Fig. 3 Attractors of a
nonautonomous, periodically forced nonlinear differential equation in regions of
quasiperiodic solutions
It is tempting to speculate that it is precisely in this situation that measures" of expansion rate or generation rate of positive entropy would serve to discriminate between quasiperiodic (entropy = 0) and strange attractor regimes, but as the oscilloscopic pictures qualitatively indicate, the bouncing back and forth between attractor regimes leads to a sequence of long transients that may be sensitive to initial conditions even if the asymptotic attractor is not. Given the well known continually changing EEG and attentional states of man and the well-established lack of stationarity of the EEG, it would seem that any local (or averaged, repeatedly local) entropic measure would be likely to always look strange without necessarily being so.
A good example of a lack of stationarity, in fact almost systematic 45-60 second oscillations in the /I> and g power spectral bands (outside of the time-scale of the usual D_{2} measurements on the EEG) is seen in figure 4. A series of 2-second power spectral transformations of the EEG called a compressed spectral array recorded 60 minutes following the administration of aminodiazopoxide, 2.5 mg/kg, to a monkey. It is seen to induce the usual b to g frequency dominance of the "valium" family of drugs, which oscillate widely over a time-scale of minutes (data from the work of Ehlers and Havsted, unpublished, 1981). This time-varying, smooth transition among dominant fast modes is consistent with a pattern of symmetry-stabilized quasiperiodicity. It is clear that what may be the very slow time-dependence of the parameter(s) of the underlying equations would lead to continually changing apparent chaotic transients, which in the time-scale of the usual EEG D_{2} or Lyapunov exponent computations would appear to reflect sensitivity to initial conditions and positive entropy generation.
Fig. 4 Example of a lack of
stationarity in the b and g power spectral bands
The relationships between the underlying attractor of dynamics at the border of quasiperiodic and chaotic dynamics and its power spectral transformation is exemplified in figure 5. It is taken from the work of Franceschini (1983) who used a five-mode truncation of a larger equation set (analogous to the three-mode truncation of the Saltzman equations by Lorenz) studied in a quasiperiodic regime. The power spectrum of the Poincarection and the flow demonstrate the characteristic broad band modes that appear in complex transformations of both quasiperiodic and chaotic dynamics, making the differential diagnosis using this transformation not possible without some other knowledge of the system. In the case of the EEG, the near F-scaling of the broad band modes suggests the presence of the most stable, irrationally related, hierarchical modes of the EEG as a quasiperiodic attractor on T^{n}.
Fig. 5 Relationship between the
underlying attractor dynamics at the border of quasiperiodic and chaotic dyamics
and its power spectral transformation - a
Fig. 5 Relationship between the
underlying attractor dynamics at the border of quasiperiodic and chaotic
dynamics and its power spectral transformation - b
Fig. 5 Relationship between the
underlying attractor dynamics at the border of quasiperiodic and chaotic
dynamics and its power spectral transformation - c
Issues of stability in T^{n} tori beyond the characteristic most irrational frequency ratio involve (as noted above) n-circular symmetries of the anatomical and functional brain. Anterior-posterior, the exchange of occipital and frontal lobe amplitude dominance when "eyes open is compared with eyes closed"; bilateral, the left-right transcallosal reciprocal facilitatory and inhibitory innervation; and vertical, "feedback" circular pathways dominate the global dynamical organization of the brain. This implies that the equations of motion (unknown) may be invariant under the relevant axial rotations and subject to their prevention of hyperbolic stability and therefore mode locking, living in a zero Lyapunov exponent state.
III. The neocortical source of the EEG signal
Although it has been estimated that less than 1 per cent of the cortical input comes from the thalamus (Dewulf 1971; White 1979), studies of EEG responses to thalamic ablations and stimulation as well as electrophysiological studies of ascending sensory pathways have emphasized the thalamus as the primary source of subcortical input to the cortex, including thalamocortical loops as the anatomical substrate of the oscillations of the EEG (the classical reference is Adrian et al. 1954). Since the characteristic times of thalamic cell discharge patterns may extend into the range of hundreds of Hz and the EEG is in Hz and 10s of Hz, it would be difficult within the context of a linear superimpositional model to ascribe the time-dependence of the EEG potentials to single thalamic cell mechanisms. However, one could quote the classical work of Cartwright and Littlewood (1945) demonstrating that a nonlinear relaxation oscillator in a quasiperiodic state (here the highly nonlinear Van der Pol differential equation) could "step down" driving frequencies by as much as two orders of magnitude. The thalamic-cortical oscillations represent an example of the vertical circular symmetry of one set of boundary conditions of the EEG modes.
It was this lack of correspondence between individual thalamic cell discharge frequencies and the frequencies of the EEG that led to a variety of other theories about the time-dependence of the EEG including the emergent properties of neural networks, global slow membrane phenomena, and, due to the suggestion of relations between spatial and temporal EEG patterns - a dispersion relation, emergent wave properties, where input is seen as perturbing an already ongoing system of vibrating strings with their own intrinsic frequencies. Cortical-cortical pathways (almost all of cortical cell input and output is intracortical) with their own natural EEG frequencies were interpreted by some to be chosen and plucked but not tuned by the subcortical input.
The thalamo-cortico-thalamic connectivity, however, leads to tuning, which occurs as a function of what would be analogous to the spatial "medium" of the waves. The wave is in effect ultrametric in that conduction occurs through heterogeneous axosynaptic pathways with a variety of conduction velocities, which vary inversely with the square of their radii. Signals are decorrelated via neuronal fibre branching with a resulting hierarchy of arrival times. In addition, neurotransmitter modulatory influences of facilitation, inhibition, and temporal dampening and delays by these "reverberating circuits" make what may be a quasiperiodic system seem more complicated. For these reasons, the shape of the EEG wave (as in physical systems) is distorted as it propagates in what is called dispersion. This phenomenon, the dispersion relation in more general form is responsible for the relationship between spatial and temporal frequencies in wave phenomena and it is known that spatial and temporal aspects of the EEG vary together with respect to frequencies, a-frequencies are faster in human subjects with smaller heads (Nunez et al. 1978). Finite propagation velocities and backward and forward conduction in the "plucked strings" of cortical wave mechanics lead to the standing and travelling wave phenomena found in the EEG when spatial and temporal frequencies are studied simultaneously (see Nunez and Katznelson 1981 for an accessible development of a wave theory of the EEG).
Spatial coherence is necessary for the generation of EEG signals of sufficient strength to monitor from the scalp since the signal from a point source such as a single cortical pyramidal cell falls off in electrical field strength as the inverse square of the distance. It is known that signals are significantly larger when recorded through holes in the calavarium from leads more directly in contact with the neocortex (for studies in man see, for examples, Heath 1954; Heath and Mickle 1960) and it is in this way that muscle artifact or other factors are not responsible for the observed spatial coherence. It is two-dimensional slabs of neocortical regions, as a layer of dipoles, which we regard as the source of the regional EEG signal. However, the intrinsic organization of the neocortex is not consonant with large two-dimensional regions of coherence in electrovoltage variation. For example, the somatosensory neocortical regions are arranged in radial, laminar columns (Mountcastle and Smith 1968; Mountcastle 1975) with predominantly local intracolumn connectivity.
It is the distribution and dynamical behaviour of the extrathalamic, brain stem neuronal input to the neocortex that helps answer both the questions of consonant time scales between single neuron discharge patterns and the characteristic EEG frequencies, and the cross-columnar, transverse temporal integration of cortical EEG activity. An understanding of the dynamics of the discharge patterns of the single neurons driving and modulating the EEG gives some insight into the quasiperiodic dynamics. It also adds to an understanding of the relations between the micro and macro scales of brain electrical dynamics of the sort sought by any statistical mechanical theory of mechanism.
In addition to the T^{n} stability arguements of maximal irrational frequency ratios and the destruction of hyperbolicity via symmetry transformations, there is a neurophysiological "reason" for the stability of the n-torus in the EEG. The neurophysiological "noise" to the neocortex is generated and transported by the neurons of the brain stem which we will treat as a source of stabilization via stochastic resonance.
IV. Hierarchical noise driving of the hierarchical modes of the EEG by brain stem neurons
In an important review of the anatomy and physiology of the brain stem biogenic amine-cell afferents to the neocortex, Foote and Morrison (1987) noted that the neocortical columnar organization was "... violated by the tangentially organized extrathalamic afferents in which single axons may innervate not only different columns within a functional region but different functional areas." A good example of the potential for global neocortical, temporal-spatial integration by one of these systems is the noradrenergic, locus coeruleus nucleus composed of a small number of cells which "...innervates every major region of the neuroaxis...." The number of cells in the nucleus grows phylogenically as does the cortex, e.g. rat, 3,000 cells; monkey, 10,000 cells; and the human, 26,000 cells. The locus coeruleus, noradrenergic system itself projects diffusely to all areas of the neocortex, demonstrates stereotyped changes in neuron activity related to EEG states, and projects extensively to another system, the reticular formation of the brain stem, whose activity is also highly correlated with the EEG state (Pickel et al. 1974; Swanson 1976; Lindvall and Bjorklund 1978; Morrison et al. 1984). The clearest example of this system's activities correlated with EEG states is the locus cells' arrest with the onset of rapid eye movement (dreaming) sleep.
A second brain stem biogenic amine system with diffuse projections to the neocortex are the serotonergic cells of the dorsal and medial raphe nuclei, which in the adult rat, are denser than those of the locus noradrenergic system. Unlike the noradrenergic system, which has some laminar and regional neocortical heterogeneity in distribution, the serotonergic system innervates the entire neocortex and is of almost uniform density across laminar layers (Lidov et al. 1980; Takeuchi and Sano 1983; O'Hearn and Molliver 1984; Kosofsky et al. 1984).
Time series of intracellularly recorded, interspike intervals, _{} (t_{i} can be viewed as a discrete first derivative, a first difference in time) from locus coeruleus noradrenergic and dorsal raphe serotonergic neurons from the work of Carlson and Foote (1988, 1989) demonstrated interesting hierarchical scaling properties within the same time-scales as the EEG. The interspike interval statistics on a characteristic neuron of each type is in the table below. The first two moments are in milliseconds. The values indicate that the locus coeruleus neuron discharges with an average rate of just under 2 Hz while the dorsal raphe serotonergic neuron on the average fires at half that rate. The general effects of both these two amine cell types on cortical cells is inhibitory, such that it could be said that the _{} of these brain stem biogenic amine neurons regulate the patterns of discharge of the spontaneously active neocortical cells (Mountcastle 1968, 1975).
Fig. 6 Time series of _{} from a characteristic locus coeruleus noradrenergic
cell
Fig. 6 Time series of _{} from a characteristic and from a dorsal raphe
serotonergic cell
Cell type |
Mean |
Var |
Skew |
Kurt |
Max |
Locus-NA |
639 |
695 |
0.063 |
2.605 |
1353 |
Raphe-serot. |
1295 |
1347 |
0.768 |
6.317 |
3611 |
The mean @ variance might indicate a Poisson distribution but the high kurtosis (reflecting the "flatness" of the probability density function) suggests that the time series of _{} are not distributed in Gaussian or Poisson manner.
Figure 6 represents a time series of _{} from a characteristic locus coeruleus noradrenergic cell (top) and from a dorsal raphe serotonergic cell (bottom). Qualitatively, it appears that the range of interspike intervals is not continuous but rather there are three or four discrete levels of values, which are visited either in runs or at least in temporal neighbourhoods. Thin lines were placed along these singular levels to suggest this hierarchical discharge pattern among interspike intervals, _{}. This impression is certainly consistent in spirit with the hierarchical scaling of the D, q, a, b, and g modes of the EEG which these brain biogenic amine neurons are theorized to be driving or perhaps less specifically, support.
We used the mean-normalized variance of distributions of partitions of systematically increasing numbers of interspike intervals (the interspike interval carrying its own metric in place of absolute time) to examine for the property of hierarchical scaling. The distribution of "box" sizes (half the distance to the next interval on each side) containing 1, 2, 4, 8, and 16 intervals. In the same way that the EEG mode lengths obeyed a F-scaling function, our hypothesis was that the distributions and their normalized variances would scale across partition size. Figure 7 portrays graphs of the density distributions of the partition sizes containing 1, 2, 4, 8, and 16 interspike intervals (from top to bottom) of the locus noradrenergic and raphe serotonergic (left and right) neurons of the table above.
The normalized variance of the partition sizes of the two systems suggest a hierarchical statistical scaling property:
Cell Type |
1 |
2 |
4 |
8 |
16 |
Locus-NA |
.586 |
.295 |
.237 |
.215 |
.210 |
Raphe-serot. |
.162 |
.136 |
.118 |
.098 |
.082 |
There appears to be a statistical scaling regime with respect to partition size in both characteristic cell types.
V. Deterministic and random models of hierarchical neuronal discharge patterns
Neuronal behaviour that jumps and sticks briefly among a finite number of allowed interspike intervals can be modelled using a one-dimensional, discrete, piece-wise convex map in co-dimension two of the sort used by Manneville as a model of an inverse saddle node bifurcation, generating an intermittent times series. Similar dynamics are found in the neighbourhood of homoclinic tangencies:
_{}mod one |
(4) |
The representative parameters of (4) are, r = nonlinear, excitatory forcing; 1/b = the recurrent, autoreceptor, inhibitory co-dimension, which controls the repolarization rate and readiness to fire, and z = its nonlinearity. Equation (4) is a one-dimensional cross-section of a diffeomorphism of the plane with neurophysiologically realistic over-determined co-dimensionality. It is not unusual for separately varying elements such as ions, peptides, and amines to act simultaneously with similar fundamental membrane actions. Post-Hopf, quasiperiodic systems which are dominated by symmetries, have a multiplicity of degenerate eigenstates, which may be seen as the abstract representation of overdetermination in the neurophysiological realm. Several variables can reach a critical value in the same state of the system.
Fig. 7 Density distributions of
the partition sizes containing 1, 2, 4, and 8 interspike intervals (from top to
bottom) of the locus noradrenergic and raphe serotonergic (left and right)
neurons
Although equation (4) does not have a singularity in the range (0,1) and it is all-over expanding, it is not hyperbolic in that the slope of the map converges on to 1 as t_{t} ® 0. This nonhyperbolicity allows smooth changes among unstable modes with the input of new sensory information to this representation of third or fourth order "noisy" neurons of the brain stem. This coincides with the nonhyperbolicity of the 0(3) symmetry-dominated n-torus of the neocortical EEG.
Figure 8 (top) is a phase portrait of equation (4) with parameter values t_{0} = 0.2, r = 0.85, and b = 3, and z = 2.5. It demonstrates four unstable fixed points representing levels of attraction-repulsion leading to _{} bouncing between their unstable manifolds. The time series (fig. 8, bottom) reflects the noisy discrete levels of t_{i} seen in the phase portraits.
Fig. 8 Phase portrait of equation
(4) demonstrating four unstable fixed points representing levels of
attraction-repulsion (top); the time series (bottom) reflects the noisy discrete
levels seen in phase portraits
Fig. 9 The time series of the
F-scaled random variable generator model (5) in
which four preferred levels of interspike intervals can be observed that
resemble the behaviour seen in the brain stem neuron series in figure 6
The random model is a generating function:
_{} |
(5) |
in which f = 1.618..., n is a randomly chosen integer from 1 to 4, and t is a random variable between 0 and 1. The same scaling law that describes the dominant quasiperiodic modes of the EEG is put into F(t), exponentiated randomly and allowed to multiplicatively operate on T as a random variable, RV.
Figure 9 portrays the time series of equation (5) with the scaling levels of f^{n} suggested as lines. Figure 10 is the probability density distributions of partition sizes containing 1, 2, 4, 8, and 16 t_{i}, respectively, of equations (4) on the left and (5) on the right.
Both the arbitrarily f-scaled random function and the deterministic intermittency map for the indicated parameters demonstrate hierarchical similarity in a crudely defined scaling regime using the mean-normalized variance of the distribution of partition sizes.
Function |
1 |
2 |
4 |
8 |
16 |
Intermit. [4] |
.385 |
.322 |
.285 |
.243 |
.185 |
Random [5] |
.599 |
.659 |
.722 |
.701 |
.731 |
VI. Stochastic resonance and quasiperiodicity in single neuron-neocortical dynamics
There has been quite a bit of interest of late in the incorporation of "noise" in electronic and computational models of single neurons and neuronal networks (Buhmann and Schulten 1987; Babcock and Westervelt 1987; Bulsara et al. 1989; Schieve et al. 1991). The simplest representation is a double-well potential with either additive (independent) or multiplicative (dependent) "noise," usually white. Sometimes, in addition, a periodic forcing term that modulates the transition probabilities is included. There is an optimal amplitude for the noise; too little and mode coherence with the periodic term does not occur, too large and the noise dominates the dynamics. This kind of time-dependent modulation of mode stability by stochastic systems, which generate sharp peaks in the power spectra, has been called "stochastic resonance" (Zhou and Moss 1989, 1990). Its application to a periodically sensory-driven noisy single neuron model has been recently explored by Bulsara et al. (1991). A weak, by itself ineffective, periodic signal is made effective, i.e. switching events occurring with the periodic input, by additive white noise. For example with respect to interspike intervals, t_{i},
_{} |
(6) |
combines additive white noise, f(t), an additive periodic term which would lead to a periodic modulation of f(t), and a bifurcation parameter r, which at critical values moves the motions of these interspike intervals from one to two values.
Fig. 10 The density distributions
of increasing partition size including (from top to bottom) 1, 2, 4, 8, and 16
interspike intervals as generated by equations (4) on the left and (5) on the
right. Note as was the case in the distribution of neuron partitions (figure 7)
that there are regimes in which the distributions appear to scale
self-similarly
An alternate formulation for a noise amplitude, mode-finding system, more directly related to the reduced Hodgkin-Huxley differential equations for membrane and neuronal activities, the Cartwright-Littlewood-Van der Pol (see figures 1 and 2) amounts to a Kolmogorov-Arnold-Moser-like stability theorem for forced-dissipative quasiperiodic systems. To paraphrase (Arnold and Avez 1968), for sufficiently small perturbations, there exists an invariant torus in the neighbourhood of the original one. For a sufficiently large amplitude perturbation, the quasiperiodic solution is perturbed to a periodic one with sharp peaks and harmonics in the power spectrum. For the differential equation at high nonlinearity, r » 1 » 1/r, in
_{} |
(7) |
b plays the role of noise amplitude, f(t) in system (6), such that "mode locking" from quasiperiodic motion can appear or not appear at the same periodic forcing frequency, w, as a function of the amplitude term b. Figure 11 is a graph plotting the (r, w) parameter regions in which periodic orbits emerge over increasing values of b. It is clear that critical values for b determine whether for a given set of values (r, w) there appear orbits of some given period. That b would be critically valued is also suggested since (higher) periodic regions almost disappear at the higher values of the amplitude.
The stability of T^{n} is governed by the ratio of the rate of contraction of orbits on to the n-torus to the exponential rate of separation of nearby orbits. The issues of symmetry, of course, as discussed above, modify the meaning of the Arnold tongues in that these would become parameter regions in which a smooth succession of quasiperiodic orbits could be passed through without hyperbolically stable mode locking.
Fig. 11 These are the results of
systematic studies on the EAI-680 analog computer of the mode locking zones
(1:1, 2:1 etc.) in black and the quasiperiodic regions in white across
increasing values of the driving amplitude term, b, in the Cartwright-Littlewood-Van der Pol system
(7). Note that varying b leads to changes
in the parameter neighbourhood of some modes, for example the elimination of the
juxtaposition of 2:1 and 3:1 at a b of
0.5. The discontinuous and irregular emergence of resonant modes across changing
b was first reported by Cartwright and
Littlewood (1945) and may be roughly analogous to the nonlinear actions of noise
amplitudes in the mode finding of stochastic resonance
Attempts to study the EEG signal parametrically usually (and almost by necessity) involve primary visual or auditory input which can engender sufficiently coherent processes as to be identifiable in both the primary sensory system nuclei and their neocortical projections. These records may involve relatively sharp power spectral modes. The neuronal theories of stochastic resonance involve the influence of noise on sharpening these observables.
The unique correlates of the EEG do not involve sensory or motor events but rather behaviourally definable states of consciousness such as day dreaming, night dreaming, anxiousness, relaxed alertness, deep sleep, and coma. The effort to "tame" the EEG signal using simple parametric manipulation on essentially peripheral sensory systems while time-locking an experimentally definable observable, misses the EEG issues of greatest import.
The neuronal systems that regulate EEG states are what Cahal has called third and fourth order neurons describing their distance from primary sensory or muscle-motor events. The theory developed here involves these third and fourth order brain stem neuronal systems involved with EEG stages and states of consciousness. In this setting, hierarchically structured noise driving by real brain stem neuronal noise sources is not seen as sharpening a coherent simple signal but as supporting quasiperiodic modes of the global neocortical EEG. The spirit of these higher order mechanisms can perhaps be viewed as both an abstraction and a generalization of the stochastic resonance model as described above.
It appears that in these global systems, both the hierarchical "signal" and the hierarchical "noise" are represented by the same brain stem neuronal sources, and that it is the neocortical T^{n}, n ³ 4, in the quasiperiodic mode that they support. The subjective-behavioural correlates of brain stem, neocortical dynamics suggest that they are systems that are generating information without input and changing states without external provocation. It is known, for example, that fluctuations in states of consciousness from light sleep to alert states occur in a wide range of characteristic times, from five-minute "cycles" to those of the day-night period. Sensory input negotiates with ongoing brain stem neuronal, neocortical dynamics rather than driving it directly. More colloquially, brain stem neuron-neocortical systems are busy thinking even without new information. We suggest that hierarchical time-scales in the brain stem neuronal noise may support the mode hierarchy of the EEG in these internal conversations.
VII. Single neuron dynamics and the EEG: Two clinical examples
An example of the relationship between single neuron dynamics and the global EEG comes from our studies of the influence on hippocampal neurons and the power spectral array of the EEG of the daily administration of lithium to rats (Mandell et al. 1983), figure 12.
Fig. 12 The result of studies of
the root mean square values of single hippocampal neuron interspike intervals in
a rat (bottom) over increasing time of sampling (each line represents an
additional minute) that show that whereas the cell without lithium pretreatment
demonstrates the typical increase in RMS with n of intermittency, the
drug-treated cell converges to a stable value within three minutes of sampling.
The compressed spectral array (top) under the same two conditions demonstrates
more widely fluctuating EEG power spectral modes in the control condition than
in the lithium-treated case, where the dominant a mode is more
statistically stable with respect to frequency
Fig. 13 Power spectral representations of the EEG in two males of differing ages. The a mode is dominant in both, though somewhat less sharp in the older subject and D and b waves are present in both. The dramatic differences are in the >40 HZ region, which we attribute to the loss with ageing of the reticular formation cells of the brain stem, which have the capacity to drive thalamocortical systems at these frequencies (Age 24)
Age 24
Age 74
Under control conditions, the root mean square fluctuations of the interspike intervals of single neuron recorded from the rat hippocampus (bottom) show the typical increase in variance with n of an intermittent system. Each division indicates an additional one minute sample of time series of interspike intervals. Lithium pretreatment demonstrated a convergence to a finite variance within three minutes of the recording. Figure 12 (top left) is a compressed spectral array of the EEG from the rat cortex demonstrating widely varying modes whereas figure 12 (top right) demonstrates a near constant a peak (for rats between 7 and 8 Hz). Lithium is known for its capacity to reduce wide variations in a variety of biological and psychological measures such that it serves to connect the micro and macro levels of brain dynamics with the same statistical change (Mandell et al. 1983).
Figure 13 demonstrates a relationship between single neuronal dynamics and the EEG by inference. It is known that significant cell loss, particularly in the brain stem reticular formation, occurs in ageing and that this has implications for neocortical driving (Mandell and Shiesinger 1990). These cells fire in the 50 to 200 Hz range. With the ansatz that single neurons of the brain stem drive and/or support a range of quasiperiodic modes in the EEG, one would predict and we observe the loss of fast frequencies above 40-50 Hz in the power spectrum (fig. 13).
VIII. Summary
Mathematical, theoretical, and neurophysiological arguments are offered to support the position that the EEG signal does not represent the observables of a strange attractor but those of an n-torus (in the vicinity of breakdown) where n ³ 4. Stability arguments for the preponderance of such a system include: (1) the presence of maximally irrational frequency ratios, shown by KAM theory to be the most resistant to mode locking; (2) 0(n) symmetry of the EEG signal in brain with n - 1 independent rotational transformations resulting in the loss of hyperbolic stability required by mode locking and stabilizing the ³ n + 1 frequency, quasiperiodic state; (3) neurophysiological systems composed of biogenic amine and reticular formation neurons in the brain stem and projecting to the neocortex that support the quasiperiodic modes with hierarchical "noise" resulting in a mode-supporting mechanism that has been called stochastic resonance.
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