J.J. Wright, R.R. Kydd, D.T.J. Liley (1993) Eeg Models: Chaotic and Linear. Psycoloquy: 4(60) Eeg Chaos (1)

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PSYCOLOQUY (ISSN 1055-0143) is sponsored by the American Psychological Association (APA).
Psycoloquy 4(60): Eeg Models: Chaotic and Linear

Target Article by Wright Kydd and Liley on EEG Chaos

J.J. Wright, R.R. Kydd, D.T.J. Liley
Department of Psychiatry and Behavioural Science,
School of Medicine, University of Auckland,
Auckland, New Zealand

jwright@ccu1.auckland.ac.nz jjw@brain.physics.swin.oz.au


Two complementary EEG models are considered. The first (Freeman 1991) predicts 40+ Hz oscillation and chaotic local dynamics. The second (Wright 1990) predicts propagating EEG waves exhibiting linear superposition, nondispersive transmission, and near-equilibrium dynamics, on the millimetric scale. Anatomical considerations indicate that these models must apply, respectively, to cortical neurons which are very asymmetrically coupled and to symmetric average couplings. Aspects of both are reconciled in a simulation which explains wave velocities, EEG harmonics, the 1/f spectrum of desynchronised EEG, and frequency-wavenumber spectra. Local dynamics can be compared to the attractor model of Amit and Tsodyks (1990) applied in conditions of highly asymmetric coupling. Nonspecific cortical afferents may confer an adiabatic energy landscape to the large-scale dynamics of cortex.


chaos, EEG simulation, electroencephalogram, linear dynamics, neocortex, network symmetry, neurodynamics, pyramidal cell, wave velocity.


1. Churchland (1986) discusses the need for the systematic arrangement of neuroscientific theories within hierarchies, so that reduction of some theories to the level of others can take place. One area requiring such reduction is that of comparisons between artificial neural networks and real neurons, including the modelling of mechanisms giving rise to the electroencephalogram (EEG).

2. We will show that correspondences exist between models for the neocortical EEG at microscopic and macroscopic levels. However, anatomical considerations indicate that while certain properties of symmetrical coupled attractor networks (ANN) are preserved on the macroscopic scale by real neurons, at a microscopic level high asymmetry of coupling appears to prevail, and the standard energy assumptions of ANN are therefore not applicable.

3. The EEG models used are the one proposed by Freeman for neocortex, and the electrocortical wave model of the authors' group, with comments embracing the global model for EEG developed by Nunez (1989).


4. In the past 25 years Freeman and his coworkers have systematically developed their model of perceptual processing in the olfactory bulb (Freeman 1964, 1972, 1975, 1979, 1987a,b, 1988; Freeman & Skarda 1985). In a recent paper (Freeman 1991) this body of information is used to make predictions concerning neocortical dynamics.

II.1 The Basic Unit of Oscillation

5. Freeman notes that pyramidal cells occur in multilayered, loosely columnar structures, with recurrent inhibition provided by inhibitory surrounds; a configuration called a KII set. Longer range transverse couplings are provided by the excitatory axonal fields of the pyramidal cells. He predicts that values similar to his estimates of neuronal parameters in the olfactory system will be found in neocortex. These include the time of transmission of dendritic depolarisation through a single neuron to near neighbours. The time of transmission is 5.8 msec., composed of 1.3 msec. of synaptic delay and 4.5 msec. due to passive electrical properties of the dendritic membrane. In conditions of sufficient excitatory drive to all the neurons in the local cortical network, oscillation will develop, imposing a fundamental frequency of 43.1 Hz upon the local field potentials (LFP) and coherence between neuronal firing and the EEG near this frequency, as has been experimentally demonstrated (Eckhorn et al. 1988; Gray et al. 1989).

II.2 Nonlinear Wave/Pulse Relations

6. Freeman also supposes that a relationship obtains in neocortex similar to that measured in the olfactory cortex (Eeckman & Freeman 1991) for the asymmetric sigmoid curve, describing the relation of pulse density fluctuation, to EEG wave amplitude. This function plots the normalised conditional pulse density (NCPD) versus the LFP emanating from the area of the neuron, and is given by

        Q = Q  (1 + exp[-(e  - 1)/Q  ])   for v > - u   ...(1)
              m                     m                o

        Q = -1 for v <= -u

where u = -ln[1 - Qmln(1 + 1/Qm)].


Q is the action potential pulse density fluctuation in the cell mass (in standard deviation units), and v is the wave amplitude of the space-averaged dendritic potential in standard deviation units. At Q = -1 all the cells are below threshold. Qm is the saturation pulse density in standard deviation units and can be considered as the system control parameter, equivalent to excitatory tone.

II.3 Locally Chaotic Dynamics

7. Freeman remarks that the asymmetric sigmoid curve and the absolute refractory time of neurons guarantees overall stability, whereas the sharp nonlinearity of neuronal threshold confers a sensitive dependence on both initial conditions and ongoing perturbation. Consequently, both the firing rate and the field potential of any pyramidal cell raised above a critical level of excitation would, according to Freeman, be chaotic. Freeman and Jakubith (1993) have embodied the principles of Freeman (1991) in two simulations which imitate the properties of small groups of interactive cells.


8. Our group has developed a model for electrocortical activity of low frequency and long wavelength (Wright & Craggs 1977, 1978; Wright 1981; Wright & Kydd 1984a,b,c; Wright et al. 1984, 1985a,b,c; Wright 1990; Wright et al. 1990a,b; Wright & Sergejew 1991), a model intended to apply to the macroscopic scale. We have first attempted to clarify the applicability or otherwise of linear wave assumptions concerning the EEG, and later to parametrise the resulting EEG theory.

III.1 Basic Assumptions

9. A unit oscillator was postulated, one considered symmetrically and reciprocally coupled to many neighbours, all described by stochastic differential equations, thus:

                       2      __
        X"  + D  X' + N  X  = \  K   X       (i != j)
         i     i  i    i  i   /   ij  j
                        D  = \  F   X
                         i   /   ij  j

where Xi is a local field potential contributed by the oscillator to the ECoG, and X'i, X"i are its first and second differentials with respect to time. All Ni, Kij, Fij, are stochastic time-varying parameters.

10. Constraints on the set of parameters {Ni,Kij,Fij} were then assumed (Wright 1990), so as to describe:

    (a) the presence or absence of couplings between units;

    (b) the effect of the reticular activating system's input of a strong
        spatially and temporally "white" signal;

    (c) special sensory inputs;

    (d) the highly nonlinear local interactions.

In particular, we assumed that because of (b) and (d), {Ni,Kij,Fij} were stochastically independent in the large.

III.2 Consequences

11. The result of these assumptions is that despite the extreme nonlinearity of the elements, the macroscopic ECoG in "desynchronisation" is predicted to exhibit dissipative linear wave motion, and equipartition of energy among resonant modes. Three tests of this theory have yielded results in favour of this hypothesis, as described in III.3 to III.5.

III.3 Equipartition of Energy in EEG Activity

12. ECoG from the posterior dorsal cortex of alert rats filtered to the 1-30 Hz band was analysed by autoregression (AR). Damping coefficients were found to be approximately equal for mode frequencies across the waveband indicating equipartition of energy among the resonant modes (Wright et al. 1990a).

III.4 Evoked Potentials Analysed as an EEG Impulse Response

13. Our theory required that autocorrelated sensory signals generate a linear impulse response in EEG activity. Using human auditory evoked response (Wright et al. 1990b) we fitted an AR model to the pre-stimulus EEG obtained one second before stimulation in each trial. At order p,

        X(t) = e(t)  -  \  a  X(t - j)                  ...(3)
                        /   j

where X(t) is the EEG, e(t) is a white noise, aj are the autoregression coefficients, and X(t - j) are antecedent values of the EEG signal to time t. With {aj} thus obtained, Equation (3) was then used as an inverse filter iterated forward in time upon the post-stimulus segment (Jazwinski 1970) so that the values of e(t) were computed for the post-stimulus epoch using information up to X(t) in the post-stimulus response. If the evoked potential is an input-following, linear oscillatory response, then the post-stimulus values of e(t) should be lag-correlated with the later evoked response, X(t + delta-t). Lag correlations of about 0.8, at 12-18 msec., were obtained for all standard channels.

III.5 Nondispersive Linear Wave Transmission

14. For damped linear nondispersive waves of long wavelength arising from transient random sources on a two dimensional plane, it can be shown (Wright & Sergejew 1991) that the signal coherence between two sufficiently close points (i.e., the case of waves arising between electrodes is excluded) is given by:

                 2      -B
                r  =  4e
                     ---------                          ...(4)
                           -B 2
                     (1 + e  )

where r2 is the squared coherence, e is the base of natural logarithms, and

                B = 2 (DR/c + SwR)/PI                   ...(5)

where D is temporal damping, R is the separation of the two points, c is the velocity of the waves, w is frequency, and S is a spatial dissipation constant.

15. Using 64 channel electrocortical recordings from the occipito-parietal dural surface of alert cats we obtained measures of average coherence over the 1-31 Hz waveband and from 0.86 to 8 mm separation of electrodes. The experimentally derived surface of r2 versus R and w was fitted to Equation (4), with D obtained from single ECoG channel AR analysis. Parameters of best fit thus yielded estimates of wave velocity, c. Notably, error residuals at best fit were so distributed as to support the presence of approximately linear and nondispersive waves.

16. A value of c of 0.1 - 0.29 m/sec. regardless of the direction of electrode alignment was found over a series of analyses. This compares to a value of 0.33 m/sec. obtained by Lopes da Silva and Storm van Leeuwen (1978) for velocity of alpha waves in dogs.


IV.1 Qualitative Comparisons

17. The abstract second order oscillator of the Wright (1990) model can be taken to correspond to the KII set in the Freeman (1991) model - that is, to the oscillating system of excitatory and inhibitory cells. Longer range cortico-cortical fibres can be equated with the Kij parameters. The stochastic independence of {Ni,Kij,Fij} is justified if interaction in local cell groups involves independently perturbed chaotic activity of high Lyapunov exponent.

IV.2 Quantitative Comparison

18. A delay of 5.8 msec. for pyramidal to pyramidal cell transmission is expected from Freeman's experiments, and our experiments imply the presence of simple linear, nondispersive travelling waves. Actually, our theory requires a complicated form of wave transmission, involving a family of nondispersive waves of differing velocity, but the mean velocity is determined by the average axonal fibre range. For waves travelling about 0.3 m/sec., the fibre range should be 1-2 mm.

19. Szentagothai (1978) gives the extent of intracortical ramification of pyramidal axons in cat visual cortex as approximately 3 mm. Other estimates of radial ramification for visual cortex of cat and primate, at 1.5-2 mm, 2-3 mm, and up to 4 mm (Crick & Asanuma 1986; Gilbert & Wiesel 1983; Rockland & Lund 1983). Braitenberg (1978) gives quantitative arguments suggesting that these axon collaterals must give synaptic contact throughout their length, so a rough estimate of the average range of axosynaptic projection is 1.5 mm. Thus fibre ranges, delay times, and estimated velocity are consistent.

IV.3 A Simulation Integrating the Two Models

IV.3.1 Outline of Simulation

20. Essential features of both the above EEG models are incorporated in the computer program schematic shown in Figure 1.


                         FIGURE 1

          Principles of simulation of a small cortical
       volume of pyramidal and stellate celles (a KII set)

        _____           _______         _____
        [ xG2]____<____ [Input]____>___ [xG1]
        [____]          [_____]         [___]
          |                               |
          |     ____    ______          __\/___________<____________
          |     [NS]_>_ [xGns]____>____ [Sum]_____<_______        |
          |     [__]    [____]          [___]            |        |
          |      |                        |              |        |
          |      |              __________|_            _|___     |
          |      |              [Pyramidal]             [xG4]     |
          |      |              [    lag  ]             [___]     |
          \/     |              [_________]              |        |
          |      |      _______<_____|                   |        |
          |      \/     [LFP]     ___|__                 |        |
          |      |      [___]     [X% T]                 |        |
          |      |                [____]                 |        |
          |      |      ________     \/                  |        |
          |      |      [Output]__<__|__________>________|        |
          |      |      [______]     |                            |
          |      |                ___|_                           |
          |      |                [xG3]                           |
          |      |                [___]                           |
          |      |     _____      ___\/                           |
          |      |____ [Gns]__>___[Sum]_____<_____                |
          |            [___]      [   ]           |               |
          |___________________>___[___]           |               |
                                    |             |               |
                                ____|_____      __|__           __|__
                                [Stellate]      [xG5]           [xG6]
                                [   lag  ]      [___]           [___]
                                [________]        |               |
                                    \/            |               |
                                  __|___          |               |
                                  [X% T]          |               |
                                  [____]          |               |
                                    \/            |               |
                                   _|___          |               |

Afferent cortico-cortical pulses provide Input. The Output is efferent cortico-cortical pulses, and the local field potential (LFP) is the observed variable. G1 and G2 are the absolute synaptic densities of cortico-cortical connections from all other KII sets. G3, 4, 5 and 6 represent synaptic densities of intracortical connections - including recurrent connections of excitatory/excitatory and inhibitory/ inhibitory type. Gns are gains of nonspecific afferents, and NS, the nonspecific afferent pulses, act as control parameter. The pyramidal and stellate lag functions are running weighted means, simulating the time-course of impact of afferent pulses upon postsynaptic axon hillocks. The fraction of cells near threshold (%T) is continuously updated from the lagged afferent pulse density.


21. A volume of cortex underlying about a square millimeter of the cortical surface was treated as a KII set, composed of KIe and KIi excitatory and inhibitory subsets. Dendritic and synaptic delay was represented by a running mean simulating the rise of postsynaptic potentials to their maximum effect upon the axon hillocks at 5 msec. from synaptic activation.

22. Threshold of activation for all cells was treated as normally distributed over the range of LFP possible between full membrane polarisation, and depolarisation, in the neuronal population. KIe and KIi subsets were coupled together by linear couplings of magnitude proportional to synaptic densities obtained from anatomical estimates, to be discussed in section VI.1. Note that recurrent couplings, from KIe and KIi sets to themselves, as well as couplings between these sets, are incorporated. The absolute magnitude of couplings was scaled to the limit of stability, beyond which scale 100 Hz oscillation was generated.

23. Within each KII set the couplings represent intracortical synaptic densities. A 20 x 20 array of KII sets were then coupled together in imitation of short cortico-cortical fibres with coupling strength again proportional to synaptic density, configured in a two dimensional normal distribution centered on each KII mass, thus linking KIe sets at longer range, in conformity with the description of Braitenberg (1978). Nonspecific cortical activation was represented in the simulation by a small continuous input delivered to all KIe and KIi sets. The simulation was run using synchronous updating at 0.1 ms intervals, under a variety of input conditions, and boundary conditions. Full results will be reported in detail elsewhere.

IV.3.2 Some Results of Simulation and Comparable Experiments

24. The simulation reproduces the 1/f spectrum of desynchronised EEG, which accords with the equipartition of energy principle. It was shown that wavefronts propagate across the neural array with velocities of about 0.6 m/sec. consistent with the upper limit of Storm van Leeuwen and Lopes da Silva's results, but 2-4 times higher than our own estimates. When sinusoidal driving of sufficient amplitude was applied to the edge of the matrix, output power was at the fundamental and integer multiples of the driving frequency, for all inputs below 40 Hz, approximating results seen in EEG during sinusoidal visual driving (EEG Journal, in submission). The spatio-temporal impulse response showed that travelling waves were not the only class of activity present. A type of local standing wave was also generated. This observation led us to apply frequency-wavenumber analysis to the simulation, and to compare the results to those from 64 channel cat electrocorticogram (ECoG). These experiments were performed in collaboration with Nunez's group.

25. The experimental recordings were obtained from the same animals used in the coherence studies described in section III.5. Both simulated and real ECoG exhibit the very surprising result that wavenumber is independent of temporal frequency (see Figure 2). The wavenumber distribution is sensitive to variation in the range of cortico-cortical couplings, and the local waves appear to arise from mutual oscillations of each KII set with its neighbours.

26. It should be noted that the travelling wave component, of velocity 0.6 m/sec., is associated with wavelengths too long to be detected by the frequency/wavenumber method using the present size of electrode array. The simulation also reproduces an asymmetric sigmoid response similar to Freeman's results (Eq. 1), and can exhibit forced local oscillation with time-varying damping, as required in equations (2).


                         FIGURE 2
        Frequency-wavenumber estimates for simulated
             and experimentally recorded ECoG

        |                    |                     |
        |                   /.\                    |
        |                   \_/                    |
        |                    |                     |
        |                   /_\                    |
        |                  //.\\                   |
        |               _  \\_//  _                |
        |        _     /_\  \_/  /_\     _         |       -1
       _|_______/.\___//.\\__|__//.\\___/.\________|_ KY mm
        |       \_/   \\_//  |  \\_//   \_/        |
        |              \_/  /_\  \_/               |
        |                  //.\\                   |
        |                  \\_//                   |
        |                   \_/                    |
        |                    |                     |
        |                   /.\                    |
        |                   \_/                    |
        |                    |                     |
        |                    |                     |
      -3.47                    -1                 3.47
                          KX mm

A very rough approximation of the frequency-wavenumber spectrum of the wave components apparent in desynchronised electrocorticogram (ECoG) measured using an 8 x 8 symmetric electrode grid (0.86 mm electrode seperation). KX and KY are wavenumber components in the X and Y coordinates of the recording array, or equivalent positions on the simulated electrocortical surface. The circles are isocontours of power. The same pattern is seen at all temporal frequencies from 1 to 30 Hz in both real and simulated ECoG. This pattern is attributable to oscillation within each KII set associated with recurrent excitation of each KII set from its neighbours. It does not reflect the presence of bidirectional (standing) waves.


IV.4 Generalisation to Global Scale

27. The method of simulation could be extended to embrace the work of Nunez (1981, 1989, 1993). Nunez's central method is to solve the following general equations in linear form. His equations are

                         /inf   /                             2
    h (r,t) = U  (r,t) + |   dv |  R (r,r ,v)g(r ,t - |r-r |)d r
     E         E         |      |   E    1      1     ------    1
                         /0     /S                      v

                         /                    2
    h (r,t) = U  (r,t) + |  R (r,r ,v)g(r ,t)d r
     I         I         |   I    1      1      1               ...(6)

where hE(r,t) and hI(r,t) are excitatory and inhibitory synaptic action densities, (i.e., the number of active synapses per unit volume at time t), and U(r,t) are excitatory and inhibitory inputs to the cortex. The distribution functions RE*g and RI*g describe the synaptic densities of cortico-cortical and short inhibitory projections connecting locations r and r1 and include the action potential velocity, v.

28. Nunez assumes synaptic action densities to be additive, and the postsynaptic membrane response to be linearisable. The general equations, and Nunez's assumptions, are completely consistent with the principle applied in the above simulation. It should also be emphasised that the Nunez approach has been applied to long range cortico-cortical connections, at a scale greater again than addressed in our simulation. At this scale much higher wave velocities are to be expected.


V.1 Standard Models

29. To introduce necessary terms some basic aspects of ANN theory are described.

    (1) Inputs to each neuron form a linear weighted sum with both
        positive and negative weight signs permitted.

    (2) Outputs of each neuron are two-state (Hopfield 1982) or graded,
        (Hopfield 1984) with a threshold value.

    (3) Connections are bidirectional and symmetric with small lags only.

    (4) A system energy which is isomorphic with that applying in an Ising
        spin-glass is defined by:

              __ __            __
        E = - \  \  w  S S  +  \   T S    (i != j)
              /  /   ij i j    /    i i
              -- --            --
              i  j             i                            ...(7)

    where E = energy, wij are the input weights, Si and Sj are the
    on/off states of the ith and jth of n neurons.

    (5) Elements in the net change state asynchronously, and the system
        vector follows a trajectory to a point attractor. In the presence
        of high asymmetry of couplings, an energy is not defined, and
        system convergence to a point attractor does not take place
        (Hopfield & Tank 1986).

V.2 Asymmetry and Chaos

30. Amit (1990) draws attention to the standard assumptions applied to these models and to their limitations in so far as they may be considered models of real neuronal cooperativity. They are generally applied with the assumption that a network of 104 elements or so is about the size of a realistic pool of completely interconnected neurons in the cortex, or elsewhere. Their dynamic properties are rather insensitive to symmetry of coupling strengths until a point is reached at which the position in state space of the basins of attraction is not fixed. Yet it may be that mobility of basins of attraction is crucial to the problem of storing and retrieving temporal sequences of inputs (Caianiello et al. 1967; Amari 1972; Little & Shaw 1975; Peretto & Niez 1986; Nebenzahl 1987; Dehaene et al. 1987). There are further advantages when asymmetric nets are considered in connection with Hebbian learning (Parisi 1986a,b; Hopfield & Tank 1986). In the presence of a "palimpsest" type of continuously operating learning and forgetting (Parisi 1986a; Nadal et al. 1986), chaotic dynamics in some synapses prevent storage while other synapses engaged in activity of a limit cycle type store memory.

V.3 Adiabatic Landscapes

31. Nebenzahl (1987) and Dehaene et al. (1987) further consider the case of symmetric couplings of two types between neurons - fast synapses, and those which begin to operate after a delay. This also confers mobility of the attractor basins, and the associated energy function (or "landscape") is said to be adiabatic (i.e., slowly changing).

V.4 Amit's Model of the Cortical Column

32. A recent neural design advanced by Amit and Tsodyks (1990, 1991) bears a resemblance to Freeman's model of organisation at minicolumn level. The Amit and Tsodyks model is composed of excitatory cells with inhibitory surrounds. The cells have an absolute refractory period and continuous membrane dynamics with realistic decay constants and rise times. The network is immersed in random continuous afferent input, as if from surrounding cell activity, with short bursts of nonrandom input as the signal to be classified.

33. These features lead to the appearance of continuous coherent intra-network dendritic potentials, usually below action potential threshold, with spikes mainly emitted because of the noisy continuous afferent. Firing rates remain low, but specific neurons spike more rapidly when the state vector is centred on an attractor. Convergence to an attractor basin is achieved by the network becoming nonergotic, but because of the ongoing noise the attractor dynamics cannot be classified as chaotic or limit cycle, any more than as point attractor.

34. Amit and coworkers (Griniasty et al. 1993), have shown that their symmetric networks can convert temporal correlations between stimuli to spatial correlations between attractors, paralleling the experimental findings of Miyashita and coworkers (Miyashita & Chang 1988; Miyashita 1988; Sakai & Miyashita 1991). The model can, in principle, be applied in conditions of coupling asymmetry (Amit, personal communication), but it is unclear at what level of asymmetry the definition of basins of attraction would be drastically affected. Notable for a parallel with Freeman's model, the Amit network exhibits oscillation with 13 ms. periodicity when membrane decay constants are in the 6-8 ms. range. Oscillation is provoked by either high excitatory tone or by persisting nonrandom afferents.


35. A further major problem must first be considered. Standard artificial network models depend upon symmetry of couplings. For want of present anatomical knowledge (Crick & Asanuma 1986; Braitenberg 1978; Braitenberg & Schuz 1991), it cannot be said that such coupling symmetry is present in the cerebral cortex. We will now describe a preliminary approach to computing cortical connectivity, following this with our further conjectures about parallels which may be drawn between cortex and attractor networks.

VI.1 Cortical Connectivity

36. To simulate realistic cortical networks, general rules in the form of equations describing the density of interaction of neurocellular components are needed. We have applied a modification of a method initiated by Uttley (1956) to compute the probability that N synapses form between a pre- and a postsynaptic cell, separated by a distance r. The assumptions made are:

    (i) the basal dendritic system of a neuron has spherical symmetry;

    (ii) the apical dendritic tree can be ignored for calculation of
    intracortical connectivity since it is involved mainly in
    cortico-cortical synapses;

    (iii) only axo-dendritic connections need be accounted for;

    (iv) an axon and dendrite meeting in a "connection space" (defined
    by the radii of intracortical axons and dendrites) form one

    (v) the probability that a particular axonal branch from one cell
    forms a synapse with a particular other cell is small.

37. Fibre density is defined as the total length of fibre originating from a given cell per unit volume. For dendrites in the visual and motor cortices, and for both stellate and pyramidal cells, Sholl (1953) has shown that dendritic density, D, is given by

        D = a exp[-r/ro]                                ...(8)

where a is about 0.002 and ro is approximately 35um.

38. Electron micrographs (Braitenberg & Schuz 1991) show that the radius of a dendrite is about 0.45um. and that of an axon is about 0.15um. Surface area of dendrites is increased by spines by about threefold, and the length of a synapse is under 1um.

39. Uttley's method, given these physical parameters, would be sufficient to determine the distribution of N if the axonal density were known in equivalent form to Equation (8). However, it is not. We have found that by assuming that fibre density for axonal trees falls off exponentially as does that of dendritic trees, but with differing space constant, a value for this space constant can be found which yields total synaptic numbers and densities in accord with those found by Braitenberg and Schuz (1991).

40. For intracortical pyramidal axonal trees and stellate cell trees, respectively, these space constants are estimated at 2.04 and 2.57 (in ro units). If, as we have assumed, connectivity between individual cells can be appropriately considered stochastic, at least at a local intracortical level, then an important conclusion can be drawn concerning these connections: THE CELL TO CELL CONNECTIONS ARE HIGHLY ASYMMETRIC.

41. Probability of connection between individual cells is shown in Figure 3. Despite the high asymmetry of connection, absolute synaptic densities of coupling between pools of cells can be considered symmetric. Such estimates were incorporated into the simulation described in section IV.3 and have been found to be essential to the reproduction of certain EEG features - the 1/f character of the desynchronised EEG spectrum in particular. Again, a detailed account of these findings will be reported elsewhere.


                           FIGURE 3
                Probability of two pyramidal cells
                sharing n synapses as a function of
                     intercellular distance

                  probability                           |
                                       -|               |\
                                        |               | \
                       -|               |               |  \
                        |               |               |   |
    1.0-|               |               |\              |   |
        |               |               | \             |   |
    0.9-|               |               |  \            |   |
        |               |               |   |           |   |
    0.8-|               |               |   |           |   |
        |               |               |   |           |   |
    0.7-|               |               |   |           |   |
        |               |               |   |           |   |
    0.6-|               |               |   |           |   |
        |               |               |   |           |   |
    0.5-|               |               |   |           |   |
        |               |               |   |           |   |
    0.4-|               |\              |   |           |   |
        |               | \             |   |           |   |
    0.3-|               |  \            |   |           |   |
        |               |   |           |   |            \  |
    0.2-|               |   |           |   |             \ |
        |\              |   |            \  |              \|\
    0.1-| \             |   |             \ |\              \ \
        |  \             \  |\             \| \              \ \
        |   |             \ | \             \  \              \|
       0-\  |\             \|  \             \  |              \
          \ | \             \   |             \ |               \
           \|  \             \  |              \|                \
          1-\   |             \ |               \                 \
             \  |              \|                \                 \
              \ |               \\                \                 \
    n        2-\|\               \\                \                 \
     u          \ \               \\                \                 \
      m          \ \               \|                \                 \
       b        3-\|                \                 \                 6
        e          \                 \                 \
         r          \                 \                4
                   4-\                 \
                      \                2
                       0                   intercellular seperation in
                                           units of ro

A diagramatic representation of the variation of intracortical synaptic coupling symmetry between two "typical" neocortical pyramidal cells as a function of their intercellular seperation. The measure of coupling symmetry (or asymmetry) is the probability that two cells share n presynaptic contacts. Intercellular distances are measured in units of ro, where ro is 31.25 um. (the space constant for the basal dendritic tree).


VI.2 Local Dynamics

42. Given that cell-to-cell connections are highly asymmetric, the energy concept of symmetric Hopfield nets would not apply to the dynamics of individual neurons. Nor would the symmetric case of the Amit and Tsodyks model be applicable. However, the combination of extreme divergence and asymmetry of couplings implies the necessary existence of many closed (and thus potentially self-exciting) loops capable of generating limit cycles. In particular, the imposition of spatially and temporally structured input could elicit resonance in closed loop pathways of specific delay, while all other possible paths continue to interact in irregular patterns. The type of learning envisaged by Parisi (1986 a,b) and Nadal et al. (1986) (see section V.2) could apply, since both locally chaotic and limit cycle activities are to be expected.

VI.3 Dynamics at Millimetric to Global Scales

43. For scales greater than the macrocolumn, up to the entire closed extent of the cortex, average couplings can be more appropriately considered symmetrical, since they are described by synaptic densities in generally bidirectional fibre pathways. Indeed, the equipartition of energy (1/f spectrum) of EEG supports this assumption.

44. From Equation (2) the instantaneous internal energy of the system can be seperated into two components that can be considered analogous to the kinetic and potential energies of a system of damped harmonic oscillators. Since {Kij,Fij,Ni} are stochastically independent in the large, in summations over high numbers of terms, their values can be replaced by their time averages. Thus

                 \            2
                 /  K  X  -  N X  - X"
           __    --  ij j     i i    i       2
K.E = 1/2 \ { -------------------------- } ...(9)
           /            __
           --           \
                        / F  X
                        -- ij j

      __ __  ____
P.E = \ \ K X X
      /  /    ij j i                                    ...(10)
      -- --

45. From given initial conditions and without further inputs the state vector would follow a trajectory of least action, viz:

                {  |  (K.E  - P.E) dt  }  --> minimum   ...(11)

until all oscillation ceased as internal energy reached zero.

46. At all points on the trajectory the system has total energy

        E = P.E  + K.E                                  ...(12)

Equation (12) can be arranged in a form similar to Hopfield's Equation (8) with the following provisos:

(i) All {Kij} (the analogs of wij) are positive; because they represent

    the excitatory couplings of cortex. Consequently the minimum at
    zero is the only energy minimum.

(ii) The K.E. has the form \ T X where T is a function of {X , X"}
                            /   i i        i                    j   i
     so the thresholds are dynamic. Slowly changing inputs to cortex,
     regulating the levels of cell depolarisation, could therefore act
     as "slow synapses."

(iii) The energy described is internal energy, not Lyapunov energy.

VI.4 Cortical Afferents and Adiabatic Control

47. Cortical-subcortical interaction is generally polysynaptic, both topographically organised and diffuse, with delays in the total cortico-fugal and centripetal extents of the interaction being considerably greater than cortico-cortical delays. It follows that cortical-subcortical-cortical circuits can be considered equivalent to slow synapses within cortex and could thus act as a form of adiabatic control of the energy landscape, since these paths act as major regulators of cell depolarisation.

VI.5 Interaction of Macroscopic and Microscopic Levels

48. Equations (9) to (12) assume stochastic independence of the time-varying parameters {Kij,Fij,Ni} and exclude consideration of fresh inputs. It is possible, at least in principle, to equate sensory inputs and transient limit cycles in local dynamics, with transient stochastic dependencies in {Kij,Fij,Ni}. Such transient dependencies would thus constitute an input term for the internal energy, separable from the stochastically independent terms assumed in Equations (9) and (10).

49. Conversely, the macroscopic level of dynamics would be reflected at cellular level as structured signals, imposed on individual cells (particularly via the apical dendritic trees of pyramidal cells) and thus activity of macroscopic scale would be capable of selectively eliciting local resonance.


50. The above considerations lead toward a view of local and global brain dynamics as a unified whole, yet they emphasise that the class of dynamics applicable is a matter of scale, as is commonly the case elsewhere in physics. A complete model would need to address the interaction between levels more explicitly, to describe in effect, how transient stochastic dependencies arise in the total cell mass. This might be addressed in simulations of individual cell behaviours. The introduction of asymmetric connectivity and a nonrandom component within the afferents "driving" the Amit and Tsodyks simulation appear likely directions for future exploration.

51. The observable EEG, in this view, is to be considered directly related to cortical information processing, and not an epiphenomenon. Conversely, much instantaneous single-cell activity may be bearing little specific information, except when self-exciting limit cycle oscillations are transiently elicited. Such transient involvement in a self-exciting loop is, we contend, the property elicited in single-cell recordings during feature-detection.

52. In these respects, our conclusions resemble those of Skarda and Freeman (1987) with respect of the olfactory system. We further contend that in the neocortex the energy concepts of symmetric ANN apply to the EEG, rather than to individual cells. This would not preclude those cortical cells which are actually symmetrically coupled behaving in accord with the model of Amit and Tsodyks, as applied to the findings of Myashita et al.

53. No contradiction of our proposal with any experimental finding is known to us, provided allowance is made for additional features of the brain, such as rhythmic driving of the cortex from subcortical sites (Steriade et al. 1990). Equally, no test of the hypothesis that is both strong and simple is apparent. Rather, it remains to be seen whether with improved anatomical estimates and detailed simulation on both macroscopic and microscopic scales, the fit to experimental data improves or deteriorates.


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