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**3 : The Diversity of Non-linear
Characteristics of the Neuron and Synapse**

**(a) The Neuron **The functional
response of a neuron is generally compared to a linear integrator
in which the firing rate is proportional to the sum of the synaptic
inputs. The purpose of this section is to illustrate how far the
neuron diverges in actuality from this simple linear model, by
investigating the varieties of non-linear behavior in the neuron
and synapse.

The integrative character of the neuron naturally leads to models in which transforms represent the integrative function. Suppose successive layers n and n+1 in a net, have a fixed serial excitation, and lateral inhibition which is normally distributed according to the function e-kx2. We can then define the following integral transform :

[3.1]

as a model in which the first term is excitation and the integral is a type of convolution between the normal distribution and changing levels of excitation. Numerical computation of the effect of this transform on the Heaviside step function, demonstrates contrast enhancement as a result of continuous lateral inhibition. A somewhat different convolving function is needed to provide easy inversion as in the Fourier transform :

and its inverse [3.2]

to model memory recall. Modification of the exponential term, through graded excitatory and inhibitory synapses, for example into an inverse Fourier transform of a step function (Crick et. al. 1988) could generate such a model. The LTM synapses of the ART configuration similarly have characteristics of mutually-inverting transforms.

Both the dynamics of excitatory and inhibitory synapses and the specific details of the transmission response display clear-cut non-linear behavior. To quote Jack, Noble & Tsien (1978) "the range of potentials over which the membrane may be assumed to be linear is very restricted". The non-linear nature of some cellular components such as the synapse are less well explored . . . "Only moderate progress has been made so far in developing mathematical techniques which might suggest ways of making an experimental determination of the electrical geometry of a particular nerve cell . . .One reason is that the the nerve cell only easily allows recording (or intracellular current passage) from one site - the cell body".

Prominent non-linearities that are well-known in the neuron are the exponentially-derived sigmoidal excitation curve [2.5], and the limit cycle resulting from the Na+ and K+ current responses to potential that is responsible for the generation of the action potential oscillation, fig 11(aii). The net current is notable in having an S-shaped curve with three zeros. The Hodgkin-Huxley equation [4.8] is specifically non-linear, as it includes power terms m3h and n4. This is extended in the Chay-Rinzel equation [4.9] to a fully chaotic equation. The response of touch receptors, fig 11(c), (Eccles 1966) illustrates the non-linear nature of excitation with membrane distortion.

Sigmoidal excitation combined with the action limit cycle results in instability, in which slight transitions from below to above threshold result in qualitative transformation from the non-firing state to the state in which at least one action potential will be emitted, fig 11(aii). Beyond the threshold, the rate response is approximately linear with input depolarizing current, fig 11(ai). Similar approximate linearity occurs over a restricted range in conversion of an action potential into a graded potential (Freeman 1983). However non-linearities enter in a variety of ways which cause both the broad periodicities of EEG rhythms and the intermittent chaos of single neurons.

Fig 11 : (ai) Piecewise-linear response of a Pyramidal
cell to depolarizing current. (aii) S-curve of current flow with
voltage results from the sum of Na+ and K+ currents. The resulting
action limit cycle has an abrupt threshold I th at which the cycle
triggers. (b) Fractal structures of superior colliculus dendrites
& the exponential decline of I & V with distance. Note
the improved current characteristics of the tree with a lower
fractal dimension. (c) Non-linear (approximately inverse quadratic)
response of touch receptors, indicating a general non-linear response
to membrane deformations.

The dendritic tree is an obvious example of a fractal structure in neurobiology (Schierwagen 1986). The growth pattern of dendritic trees has been successfully described by a diffusion-limited aggregation model (Sander 1987), common to several fractal deposition processes. Differing fractal dimensions ranging from 2.1 through to 2.7 result from the diverse anatomies of different cell types. A lower dimension results in fatter dendrites and a higher current transfer, fig 11(b). The continuous dendrite voltage declines exponentially with distance, and the current shows a related fall-off. The combination of excitatory and inhibitory influences is thus spatially dependent, with some synapses occupying strategic positions at the base of particular dendrites. Dendritic and synaptic growth and atrophy with learning cause long-term dynamic change. In the 'twitching spines' concept (Crick 1982), changes in the spines may occur over milliseconds.

The structure of a pyramidal cell neuron is illustrated in fig 12(a) (Eccles 1966). The total number of excitatory and inhibitory synapses ranges up to 200,000 and covers a variety of regions and structures. Although the neuron clearly has the characteristics of integrating the positive and negative influences of hyperpolarizing (inhibitory) and depolarizing (excitatory) synapses, this function is complicated by the variety of physical types of synaptic structures in a complex neuron, the variety of synaptic contacts made by a single cell with several other cell types, and the varying positions of key synapses with respect to specific dendritic branches and the time-dependent changes accompanying facilitation and learning. Dendo-dentritic connections and synaptic microcircuits make it possible that the neuron is essentially a super-unit carrying out multiple processing tasks, each with non-linear potentialities.

A second striking aspect of the neuron is that the membrane is a two-dimensional electronic surface capable of different connectivity properties from one-dimensional systems. Although the neuron is conceived of as an integrative processing unit, both the synaptic junction and the cell body form independent functional sub-units. The cell body both selectively integrates and also often bifurcates the input between active and passive states through oscillatory instabilities at threshold. Growth of new synaptic spines and rearrangements of specific dentritic trees has been associated with learning and in particular, PKC-mediated memory formation (Alkon 1989, Mishkin & Appenzeller 1988). This may occur in local synaptic microcircuits through the cell body making a sub-activation response to a local associative synapse pair as illustrated in fig 12(b) (Alkon 1989).

Fig 12 (a) Structure of a pyramidal cell (Eccles
1966) showing the diversity of synaptic types on a single neuron.
(b) Local fixation of a conditioned response (Alkon 1989) involving
dendritic microcircuitry.

The neuron is an amoeboid cell capable of exploratory changes in its dendritic and synaptic linkages which go beyond simple additivity. In at least some species growth of new neurons takes place (Nottebohm 1989). The dynamics of the neuron is complemented by cells such as the oligodenrocytes, which mylenate the axons and the astrocytes which surround neurons in the brain & blood-brain barrier, buffer external K+ concentration, service synaptic molecules, and may have a role in long-term potential changes (Kimmelberg & Norenberg 1989).

Non-linear aspects of neuronal conduction are essential in the following:

(1) Discrete decision-making events. Passage past threshold signals "an event" in digital form, causing the unstable bifurcation of continuously varying input into active and passive states.

(2) The stability-breaking of mutual (dipole) inhibition in layered nets.

(3) The development of complex time-dependent dynamics as a consequence of non-linear feedback interactions, for example in loop circuits displaying sensitivity to external input through chaotic excitability.

Neurons use at least two means to communicate and transfer electrical states from one cell to another. In the electro-electric junction membrane potential is transferred directly, whereas in the electrochemical junction a transduction takes place, involving initial release of neurotransmitting chemicals from the presynaptic membrane, binding of neurotransmitters to receptor sites on the post-synaptic membrane, and subsequent hyper- or de-polarization of the post synaptic membrane. This occurs either by direct effects of the binding protein as an ion channel, or indirectly through the action of second signalling molecules within the cell such as cyclic AMP, which activate a protein kinase which blocks K+ flow, thus prolonging depolarization and hence increasing Ca++ inflow (Rasmussen 1989).

Synaptic transduction involves several steps with regulatory feedback. The neurotransmitter is generally stored in membranous capsules called synaptic vesicles similar to those produced by the Golgi apparatus. Depolarization of the synaptic membrane involves both Na+ and Ca++ channels. The calcium ions promote the binding of the synaptic vesicles to specific receptors on the inside of the pre-synaptic membrane and consequent release of the neurotransmitter into the synaptic cleft. Here the neurotransmitter may bind to an receptor site in the postsynaptic membrane, or to presynaptic receptors which provide feedback control, or be assimilated by enzymes such as monoamine oxidase, which remove surplus neurotransmitters after stimulation, fig 13(a).

As an example, the acetyl-choline receptor at the neuromuscular junction requires two acetyl-choline molecules to activate it (Stevens 1979, Darnell et. al. 1986). A single synaptic vesicle with 10,000 acetyl-choline molecules will activate about 2000 sodium channels, taking about 100 microseconds to traverse the synaptic cleft. Binding of the transmitter lowers the energetic stability of the non-conducting channel resulting in a conformational shift to the open state. After a random interval this will switch to an inactive resting state and finally back to the active resting state when the membrane potential has risen. Such channels may stay open for anything from one millisecond to several hundred depending on the specific type of synapse. About 20,000 ions can traverse an ion channel during the time it is open. Only about 1 in 103 ions is required to traverse the membrane to shift the potential through the action range. A single synaptic vesicle at the neuro-muscular junction reduces the post-synaptic potential by about 1 mV. Although this is too small to reach threshold, it shows on a recording as a quantized potential spike. By contrast, it has been postulated that some central nervous synapses may function at levels where the order of only one vesicle released will communicate an active state across the synapse, eliciting quantal information transfer.

Fig 13 : (a) Neurotransmitter pathways & the
synapto-synaptic junction illustrate multiple feedback links and
bilinear responses. (b) Marr's model of the cerebellum, showing
inhibitory interneuron connection and threshold tuning. [g-granule,
go-Golgi, s-stellate, b-basket, p-Purkinje, mf-mossy fibre, pf-parallel
fibre, cf-climbing fibre].

In the standard model, neurotransmitter release is quantized through exocytosis of synaptic vesicles. However some neurotransmitters such as acetyl-choline (Cooper et. al. 1982) may also be released without exclusive use of vesicles. The principal action of neurotransmitters is facilitated by binding to receptor sites, which even for a single neurotransmitter such as acetyl-choline may bind to both excitatory (e.g. nicotinic) and inhibitory (e.g. muscarinic) receptors. However the common involvement of choline in both phospholipids and neurotransmitters, combined with the common structures of serotonin & the catecholamines, both of which are derived from aromatic amino acids by specific amine and OH substitutions, suggests that these primitive molecules may have been selected through intrinsic chemical interactions with the non-polar and ionic components of the primitive lipid membrane. The selectivity for amine derivatives may be based, for example, on interactions with primitive ion channels, in which the amine moiety serves to stabilize flow of positive ions (King 1990, Mueller & Rudin 1968).

Equations such as [3.3]

display first-order kinetics if a single copy of a given reactant such as R1 enters the kinetic interaction, as opposed to the two copies of R2. However key steps in conversion of excitation, such as neurotransmitter binding to ion channels, require two molecules of neurotransmitter per channel and hence behave quadratically like R2.

The complexity of synaptic kinetics including presynaptic receptors, lytic enzymes, synapto-synaptic and dendro-dendritic junctions, and microcircuits provide further sources of non-linearity. In fig 13(a) is shown a synapto-synaptic junction in which a neurotransmitter such as serotonin facilitates the sensitivity of an underlying synaptic junction by binding to cyclic AMP producing sites on the main synapse. This subsequently increases the Ca++ concentration and makes the second synapse more sensitive, through making the vesicles more accessible to the surface receptors that promote their emptying into the synaptic cleft. Such synapto-synaptic junctions result in bilinear response characteristics over time intervals, because the post-synaptic potential is a product of pre-synaptic and synaptic inputs [3.4]

Such triple junctions may be important in the production of learned responses. Simple nervous systems, such as those in the sea snail Aplysia (Kandell 1979), have synapto-synaptic junctions which appear to function in facilitation in learning through this multiplicative effect. Note that the Lorentz system fig 1(a) is purely bilinear.

(4) Evidence for Chaos and Fractal Dynamics in Excitable Cells.

**(a) Membrane electrodynamics
:** The Hodgkin-Huxley model of (1952) of the action potential
is the foundation of modelling of the dynamical response of excitable
cells, and as such has led to the development of the Chay-Rinzel
model of the chaotic neuron. To derive the model, we assume that
the ionic current for each principal ion i = K, Na is
:
[4.1]

where gi is a function depending on the proportion ni of ions inside the membrane, V is the displacement from resting potential and Vi is the standard potential of the ion i.. Generally, the rate of change of ni is given by [4.2]*

where a(ni) and b(ni) are the rate constants for the inward and outward movements of ions for a given ni with the proportion outside being 1 - ni. Since at equilibrium for any given voltage, we havewe can define steady state ,

where [4.3]

Let be the equilibrium states before and after clamping with a voltage V, then

[4.4]

The inverse relations to [4.3] [4.5]

can then be used to calculate experimentally as a function of voltage, returning, for example, in the case of K [4.6]*

If a given ion is transported through binding to the channel in k-tuples, the ionic conductances will obey the power law for some power k of ni. [4.7]

This can then be put together into a total (capacitative plus ionic) current equation (h = Na inhibitor, L = leakage):[4.8]*

The asterisked equations [4.2], [4.6] & [4.8] in combination then form the Hodgkin-Huxley system.

Fig 14 : (a) Clumping in successive periods in
ECG is indicative of chaos rather than noise, (Babloyantz 1989).
The figure plots successive delays, demonstrating a non-random
spread. (b) The form of g(q) in chicken heart cells, from a qn-
qn+1 plot. This closely approximates the form of the circle map
function fig 5(e) justifying the use of the circle map in the
model. (c) Periods in chicken heart aggregates used to develop
the model in detail. (d) Period 3 attractor in Nitella,& interpolation
of the transfer function by taking repeated iterates and plotting
Vn+1 against Vn to form the analogue of the figures in fig 4(b)
for this map. (e) Forms of chaotic waveforms in the Chay-Rinzel
model (bursting) & in Nitella. (f) period doubling in Chay-Rinzel
model (beating) shown by plots of V, Cai and t together.

It is well known that a variety of excitable cells including in particular the b-cells of the pancreatic islets display action potentials which are capable of both chaotic bursting and a variety of periodic modes of activity. This is caused by the action of glucose lowering intra-cellular Ca++ and appears to involve four components, Ca++ and V activated K+ channels, a V activated Ca++ channel and effects of glucose on Ca++ via ATP. Chay & Rinzel (1985) successfully adapted the Hodgkin-Huxley system by replacing the Na term with an identical Ca term and including

the Ca activated K channel in : [4.9]

and an extra term [4.10]

representing the removal of internal Ca by pumping and its inflow via the Ca channel.

The dynamical behavior of the model compares closely with the actual activity of such cells and includes in particular a series of regimes as the rate constant for removal of calcium kCa is varied upward, including periodic & then chaotic bursts, followed by chaotic & then periodic beating. The periodic beating arises through the period doubling route as in fig 14(f). Since Ca++ is also central in synaptic exchange, such dynamics may generalize to neuronal networks. Similar experimental results occur in neurons. Sinusoidal stimulation of the internodal cells of Nitella result in the formation of period 3 bursting and hence chaos (Hayashi et. al. 1982), fig 14(a), and deterministic chaos has also been defined in the activity of single neurons. Enzyme systems such as the glycolytic pathway have likewise displayed chaotic and period 3 dynamics, both in experimental models and in yeast cells under sinusoidal glucose concentrations (Markus et. al. 1985).

The circle map is useful in the modelling of chicken heart cell aggregates, which are stimulated by a periodic pulse (Schuster 1986), figs 14(b), 5(d). Like the dynamics of the cortical EEG, the phase shift of the stimulus is the determining parameter in the dynamics. If the natural period is t, the stimulated period is T and d is the stimulus delay, we find that the function :

[4.11]

has form equivalent to the circle map, and by examining a single wave repeatedly pulsed, we get from fig 14(c) [4.12]

so dividing by t , [4.13]

where ts is the interstimulus distance. Such cells display mode-locked beating and chaotic changes as ts is varied.

Periodic stimulation of giant squid axons have similarly demonstrated mode-locking, bifurcation to chaos and the transmission of chaotic signals along the nerve axon. Spontaneous chaotic firing was also demonstrated in the mollusc Onchidium under synaptic blocking by Co2+ ions (Hayashi & Ishizuka 1986), and in simian motor neurons (Rapp et. al. 1985), and single cortical cells (Albano et. al. 1986a) establishing the involvement of chaos in individual neurons in vivo. Neuronal networks which involve the interaction of many resonant systems thus have a high probability of displaying chaotic dynamics through the quasi-periodicity route.

**(b) Fractal Dynamics in a Voltage-dependent
K+ Ion Channel.** The traditional model for ion channel kinetics
is that the channel has open & closed states, each of which
has a rate constant

Given the cumulative probability P(t) that a channel will remain closed over [0,t], we have

[4.14]

Hence [4.15]

and so [4.16]

Hence [4.17]

In a Markov process k0 is constant as in fig 15a[1] and we get : [4.18]

If we refine the Markov model to include further discrete states, such as closed-closed-open, we get two exponentials and a step curve for k0(t) as in a[2]. If instead we look for a power law of t (Liebovitch et. al. 1987 a,b) : [4.19]

as in the fractal equation [1.16], the log-log plot will give a sloping line as in a[3], and P(t) can be calculated as an integral from [4.17] and [4.19]. The experimental plots in fig 15(b) show a fractal model gives the best fit for ion channels (Dc= 0.79) in the corneal endothelium and also in hippocampal neurons (Dc=1.07, Do=0.34).

Fig 15 (a) Theoretical predictions of Markov &
fractal ion channel models. Note the Markov models give horizontal
or step functions, requiring an infinite number of states to represent
the same effect as the fractal model sloping line. (b) The K+
channel kinetics in a corneal endothelium cell, confirming conforms
to the fractal model. (c) Relaxation of excited myoglobin involves
four functionally important motions and many equilibrium fluctuations
illustrating the basis of the fractal model in diverse molecular
excitations. (d) The 3-D pathways of relaxation of the myoglobin
molecule.

The significance of this can be best understood by looking at a better characterized system, photo-dissociation of CO-myoglobin (Ansari et. al. 1985). In fig 15(d) is shown the structure of myoglobin illustrating how a global conformational reaction occurs to the relaxation of the dissociated molecule. This relaxation involves a hierarchy of several functionally important motions (fim) each with several equilibrium fluctuations (EF) having varying potential barriers and rate constants. Relaxation thus involves many quantum interactions in which phonons, Davydov solitons, coherent and other excitations are exchanged. Various electronic orbital structures such as delocalized electrons, spin-orbit coupling, triplet-state electrons and dipole structures which could enable Bose condensation of coherent photons have been proposed in biomolecular systems which profoundly complicate this fractal structure. Frölich (1968,1975) has suggested that the cyto-skeleton and membrane may admit coherent states. The complexity of three-dimensional structure of a protein, resulting from the hierarchy of strong and weak bonding & water structures, and the many conformational changes possible make for a high-order of variation in state on many scales consistent with the fractal model. Aperiodicity in protein primary sequence also favors a cascade of distinct interactive quanta in tertiary structure. Such considerations apply alike to the proteins including enzymes, ion channels & microfilaments and also to the piezo-electric structure of the membrane.

**(5) Chaos and Chaotic Models in Neurosystems**

**(a) The Freeman-Skarda model
of the Olfactory Bulb**

Although the general idea of chaos as a foundation for brain dynamics has been mentioned by several researchers over a considerable period (Nicolis 1983, 1986), the model of Skarda & Freeman (1987) is one of the most completely developed from both a theoretical and an experimental point of view. In this model, chaos is proposed as the basic form of collective neural activity for all perceptual processes, whose function is threefold : as a controlled source of noise, to access previously learned patterns and to learn new patterns. Although computer-based information processing concepts have been widely used in modelling peripheral and some central sensory systems, the specific structures of feature-detection and command neurons have not been successfully extended to the associative areas of the cortex. Skarda and Freeman comment "brains don't work in the way everyone including ourselves expected them to. The form in which sensory information is represented in the olfactory bulb is a spatial pattern of chaotic activity covering the entire bulb involving equally all the neurons in it and existing as a carrier wave or wave packet".

Fig 16 (a) 8x8 matrices of bulb response illustrating
topological differences in excitation for air and amyl nitrate.
(b) Modular nerve cell assembly, connecting the olfactory bulb,
(OB) pyriform cortex (PC) and entorrhinal cortex (EC) involves
mutual inhibition (e.g. GG, II, BB) and broad feedback pathways
(L3 etc.) (c) The realization of the above model circuit in a
network of bulb cell types. (d) EEG in seizure in the bulb and
model simulation, showing the dynamics of the model agrees with
experiment. (e) Bifurcation diagram and (f) Phase portrait of
the stages of chaotic excitation in the model.

On inhalation, a transition occurs from low level chaos fig 16(f) to a trajectory which in the case of a familiar odor will settle into one of several periodic orbits, but in the case of a new odor will avoid existing periodic attractors, hunting chaotically until a new periodic attractor is established over time, forming both a new familiarized response and a new symbol. Inhalation thus causes a bifurcation of the dynamic into one of many periodic orbits for each learned odour, embedded in a chaotic regime. An unusual odor results in a low peak frequency and broad spectrum with excessive phase modulation. Similarly the work of Skinner et. al. (1989) has demonstrated that novel stimuli increase the correlation dimension in the bulb, while familiar stimuli reduce it. The dynamics proposed consists of two modes. In the diastolic mode the bulb responds to input from afferent neurons - (broad spectrum states). In the systolic, the bulb goes into high-energy interactive bifurcation. This two-phase activity is remarkably similar to the two-phase activity of the ART architectures.

The chaotic dynamic may provide a way of entraining neurons that is guaranteed not to lead to cyclic or spatially pre-structured activity, allows unbiased access to every limit cycle attractor, provides escape from established attractors under unfamiliar stimulus and ability to create new limit cycle. The response has extremes of seizure and coma representing a chaotic attractor and fixed attractor. The 'background' odor likewise consists of low amplitude chaotic activity, which on measurement gave a correlation between 4 & 7. The dynamic becomes latent during exhalation and in the absence of motivation. These ideas are illustrated in fig 16(e,f) .

An experimental program was carried out in which a conditioned odour stimulus CS elicited a conditioned response CR indicating discrimination ('licking' only in the case of CS+). A 3.5 x 3.5 mm 8 x 8 array of electrodes was implanted with a 2 msec sampling frequency. Filters were set at 10 and 160 Hz. A 76 msec sampling interval thus required 64 x 38 x 12 bits. The 64 traces were fitted by regression to 5 elementary basis functions, and a residual variance from the sum of these, and the high and low filter matrices were taken and tested for capacity to classify events correctly wrt CS and CR. Although the raw EEGs did not show obvious distinguishing characteristics, the first basis matrix was found to be an effective measure of discrimination, once occasional artifacts were removed. A sample amplitude matrix is illustrated in fig 16(a). Chaotic dynamics is likely to result from an interplay between the major neurotransmitter types such as GABA and nor-epinephrine underlying the specific classes of neuron.

A 4-module nerve cell assembly model is illustrated in single component form in fig 16(b). The (P)eriglomerular cells are excitatory to each other and to (M)itral cells. Inhibitory interneurons [(g)ranule etc.] are self-inhibitory and mutually excitatory among mitral cells in the bulb and pyramidal cells in pre-pyriform cortex. A more detailed neural circuitry for the bulb is shown in (c). A simulation of the NCA structure, based on second-order non-linear differential equations for each neuron was carried out by varying the excitatory gain between P and M. This resulted in a transition from quasi-periodicity to chaos and gave output similar in form to states of the EEG as in fig 16(d).

The overall dynamical nature of CNS processes may involve chaos as a fundamental component because resolution of one aspect of the dynamic into a stable attractor may result in destabilization of subsequent centres. For example convergence in the bulb may destabilize motor centres involved in the reaction to the stimulus. The stabilized behavioral motor action (licking) then leads to further input. The dynamics thus depends on constraints of self-organization through arousal and motivation. This results in time-dependent [non-stationary] dynamics in which attractors arise and decay through bifurcations. Furthermore, real behaviors display both multifunctionality of responses and the blending of several repertoires in a single behavior (Mpitsos 1989). For example in the mollusc Pleurobranchaea feeding, exploration or regurgitation actions may be combined. This poses a serious problem both for the experimentalist trying to establish a dimension from a time series and for the theorist attempting a description of the attractors or their resulting behavior. The nature of the dynamics may consist of a fractal domain in which attractors occur on differing time scales. The basis of an attractor is thus unlikely to lie in rigid anatomical structures, but in dynamical interaction. Chaos also involves structural instability as specific classes of neurons are bought to threshold and undergo an exponential response with depolarization.

Like the connectionist models of neural nets, the Skarda-Freeman model utilizes parallel distributed processes to lead to a self-organized optimization. However, most neural net models are digital or seek local minima and do not admit chaotic background states. The connectionist models do not generally involve the local feedback necessary to develop chaotic and limit cycle dynamics. For example the Hopfield net has point attractors and does not lend itself to dynamical refreshing. The inherent problems of entrapment at local minima in neural nets are handled in the Boltzman machine by stochastic annealing rather than chaos. Back propagation as a process is slow and dispersed in the brain.

A variety of possible functional roles have been suggested for chaotic processes :

(1) they allow unbiased access to stable states, provide escape from stable states under unfamiliar stimulus and the ability to create new states.

(2) they provide a basis for the development of symbols as the stable attractors of the dynamic from a more fundamental dynamical continuum. Thus while the models of artificial intelligence cannot fully represent chaos, dynamics may be able to represent symbol creation and manipulation.

(3) they provide a natural spatially-global basis for developing self-organization through stability structures, similar to the role of protein tertiary structure in complementing the data storage of the genetic code. Physical systems such as turbulent fluids display formation of fractal dissipative bifurcations, which are a rich source of new structure. Chaotic variation may thus be a central route for developing new structures in the brain.

(4) the need for random processes such as annealing, provides a basis for chaotic fluctuations to escape local minima in constrained optimizations.

(5) they may enable a very efficient form of data compression in memory in which the complexity of a given CNS state is reduced to the topological form of key attractors. A similar role has been proposed for the filtering action of integrative attention.

(6) they provide for a potentially indeterminate brain consistent with the roles of consciousness and free-will.

Critics of such models claim that it has not been demonstrated why chaos is essential to be able to carry out the tasks required in the bulb or C.N.S (Skarda & Freeman 1987). Other mechanisms such as the ART model, are cited as alternative explanations which can 'self-organize, stabilize, & scale a sensory recognition code in response to a list of binary input patterns'. The matrices in the experiment are claimed to represent more the state of motivation and the adaptive response than pattern recognition, despite being pre-sensory. Indeed relearning a previous odor after aversive conditioning results in a new pattern. However according to Freeman, adaptive resonance models based on mismatch detectors which stimulate arousal if match is insufficient, do not appear to have supporting structures in the neuronal architecture, because feedback from the pre-pyriform cortex to the bulb is not topographical. Nevertheless the obvious similarities between the functional two-phase modes of operation of each suggests compatibility of the theories. It is possible that further investigation of such systems will demonstrate a complementation of these two models in which plasticity in feature detection requires a continuous unstable dynamical version of the ART configuration in which dynamical chaos is identified with the sequential search phase.

**(b) Experimental Chaos in the
Electroencephalogram**

The status of the EEG, despite for a time being eclipsed by that of single neuron recordings, has since become a classic area for investigation of chaotic dynamics at the neurosystems level. Although the EEG is described as periodic and a series of frequency regimes have been defined [ 1-3 Hz d, 4-7 g, 8-14 a, >14 b ] (Lane 1986) its power spectrum has a broad band spread characteristic of filtered noise or chaos, fig 17(e). The source of the EEG remains obscure. Although it was initially proposed to arise from collective action potentials e.g. from pyramidal cells, the lack of correlation between single cell recordings and the EEG has led to two hypotheses, local variations in dendritic potentials and changes mediated by K+ transport in glial cells, as supported by data from electroretinograms (Galambos 1989). It may involve all of these in varying measure. Bullock (1989) stresses the importance of cooperativity in generating the micro-EEG over short time (10-1s) and space (1 mm) intervals and the correlation between both unit spikes and graded potentials in a fair sample of the neural population.

The obscurity of the EEG raises the question as to whether it represents functional or maintenance activity, however the relation, firstly between the EEG and broad mental states, and secondly between evoked or event-related potentials (EPs or ERPs) and behavior support the relevance of the study of the EEG to the form of brain dynamics. ERPs such as the 'expectancy' wave & P300 (Hooper & Teresi 1986, Maurier 1989) are well-known. In fig 17(d), a smaller evoked potential amplitude occurs for a preceding higher amplitude EEG, and in 16(c) anticipated stimuli result in coherent EEGs (Basar et. al. 1989).

Once signal averagers made possible the extraction of evoked potentials, researchers such as Basar (1983a,1990) began to suggest the possibility of EEG fluctuations being a result of a strange attractor. Subsequently, Babloyantz et. al. (1985,1986), Rapp (1985) and other researchers established low-dimensional attractors in Slow Wave Sleep, epilepsy and other brain states, fig 17(a) (Babloyantz 1989). The occurrence of low-dimensional attractors, in both pathological and natural states supports the involvement of chaos as a fundamental aspect of brain dynamics.

The higher dimensionality of the active waking brain, fig 17(a), is consistent both with high-dimensional chaos and with filtered noise. Models have been proposed using coupled linear oscillations with noise (Wright 1989,1990) for reticular activation. The brain may in fact utilize all three modes : chaotic, periodic and stochastic. Some discrete forms of information processing for example could appear as noise. The activity of single neurons has been found to include both cells with activity indistinguishable from noise and also neurons displaying low dimensional chaos (Albano et. al. 1986a). The development better techniques for discriminating quasi-periodic noisy data from chaos in higher dimensions than 7 is essential before any detailed analysis can be made into the structure and variety of global neurosystems dynamics. However the status of chaos as an aspect of neurosystems dynamics appears well-confirmed.

Fig 17 : (a) Typical correlation dimensions of
a variety of natural and pathological brain states. (b) A 2-D
map of a chaotic attractor in stage 4 sleep (c) Coherence of EEG
recording in anticipated events and desynchronization in the absence
of anticipation. (d) Average Evoked Potential is reduced for a
prior larger EEG. (e) Time-evolving 2-D power spectra of a-rhythm,
occipital and frontal showing variation with time in both the
frequency spectrum and the correlation dimension as well as significant
differences in the dimension between frontal and occipital dynamics.

Although the discovery of chaos has been identified with the lower dimensional states such as sleep and pathological states, higher dimensions do not contradict the existence of chaotic dynamics. It is thus wrong for researchers to conclude that chaos is only an attribute of the quiescent or malfunctioning brain. Thus low dimensions have sometimes been associated with a more maintenance-oriented activity and high dimensions with noise or varying information content. Very high dimensions consistent with noise or chaos could still result from complex chaotic dynamics with a vast number of interacting subsystems. Dimensions below 10 are particularly significant when the very large number of degrees of freedom in a system comprising 10^{10} neurons and 10^{15} synapses is considered, and high dimension or non-saturation may reflect multiple attractors rather than random activity per se, such as spontaneous background firing. The structural stability of periodic attractors make them insensitive to perturbations such as external stimuli. Chaotic dynamics is thus suggested as having a role of enabling varied response to stimuli, providing the capacity to adopt new states outside a fixed repertoire and providing for an increasing proportion of new information in the system in relation to the initial information - i.e. responsiveness.

Studies of the a-rhythm illustrate the complexity inherent in studying chaos in the EEG. The correlation dimension varies from 8 down to between 3 & 5 depending on experimental techniques such as 5 - 15 Hz filtering. The dimension will vary both from time to time and from electrode to electrode, frontal areas often having high dimensional activity which does not reach saturation of embedding dimension and occipital areas having low dimensional chaos, for example with eyes closed. Individual local cells groups may or may not be either blocked by opening the eyes, driven by flicker or altered by specific mental tasks. It thus appears that specific strange attractors may be dynamically variable and subject to creation and destruction by bifurcations. Bullock has drawn attention to the fact that a well defined saturation plateau indicates high determinacy in the signal with low noise. However distinguishing dominating resonances such as the occipital a-rhythm from less energetic components needs further analysis. Time-evolution of the power spectrum, fig 17(e) is one way of extending the analysis (Basar 1990). The variation of waveform with brain area and overall change in correlation dimension over time are both evident.

Several experiments support event-related desynchronization of a activity. In Basar et. al. (1989), anticipation of a repetitious omission of a regular stimulus showed coherence in EEG between successive sweeps, while lack of certainty induced by random omissions resulted in desynchronized waves, fig 17(c). Hoke et. al. (1989) using magnetoencephalography demonstrate elegantly the development of coherence in the event-related MEG upon the occurrence of a repeated signal. In fig 18(a) is shown the development of two regions of coherent phase. Independence of variation of mean amplitude from phase changes in some of these steps indicates distinct mechanisms may be involved. These results further support the idea of the chaotic epochs in a dynamic being the ones in which sensitivity to changing input occurs.

Higher frequency activity has also reported, including 40 - 60 Hz oscillations (Basar et. al. 1989). Evidence supporting coherence at these frequencies between functional columns has been found in the visual cortex, suggesting that resonances rather than specific neuronal architecture may be the substrate for specific perceptual events. Even higher frequency activity 100-1000 Hz occurs in the cerebellum and brain stem centres. Major resonances, such as a and q, may thus result from resonances in the connection of cortical, thalamic and brain stem centres, despite the pacemaking periodicities e.g. of a apparently being driven by the thalamic nuclei.

The existence of dominant frequency bands supports the notion that there exist periodic oscillations or predominant low-dimensional attractors with intermittent resonance, which maintain coherence in neural networks. Note from fig 2(a) that chaotic systems can also have dominant bands. Intermittency may be essential to avoid resonances preventing responsiveness or even reaching seizure. Resonance between lower level neurons activated by the same stimulus would enhance the input to appropriate higher-level cells. If the resonant frequency of a cell varies with its hyper- or de-polarization, cell populations will enter or leave the resonance depending on their synaptic input state. Models have been developed for major oscillations based on time-dependent coupled linear oscillators. These do not however model the chaotic feedback aspects of dynamics.

Other models based on chaotic oscillators which intermittently excite specific neuron sub-assemblies have also been proposed. These highlight another very different possible role for chaos, the compression of input data (Nicolis 1983,1986), Skinner et. al. (1989)). A governing attractor system which forms a dynamic model of the input, contains far less data than required to describe all the dynamical states, including their fractal boundaries and hence permits a very significant data compression. Such mechanisms also provide a role for suppression of familiar input (e.g. in the thalamic a-oscillator), selective attention having the function of simulating only the unfamiliar and hence non-simulated aspects of stimuli. This mechanism is also reminiscent of ART. Note that chaotic PLMIs may also form periodic nets (Labos 1986).

The dynamical model requires development of a scheme which possesses modulated resonances or modulated coupling between cells or cell-assemblies in the layered neuronal network. A basis for such a threshold-feedback model is a layered net in which the internal feedback is modulated by altering the threshold of the interneurons responsible for maintaining the feedback between cells within a layer. As the mode moves from systole to diastole a change in these thresholds would convert the layer from positive or neutral (linear) coupling, similar to a stable laser, to unstable equilibrium through mutual negative feedback, resulting in structural instability and generalized competitive bifurcation, or to asymmetric feedback permitting limit cycles and chaos. On a modular scale this could direct selective attention to unresolved features. The circuit of fig 16(b) has elements of this structure through the feedback between the 4 modules and the excitation and inhibition within the net of a single module. A key element in the model is identifying the synapses mediating the threshold change responsible for mode change. The major cortico-thalamic-basal resonances mediating arousal in the CNS could thus be based on the modulations which hold the stable attractors, but systematically perturb the fractal regions of chaos until a major attracting system emerges, forming a dynamical version of ART.

Fig 18 : (a) Spatial coherence in an Event-related
Potential (Hoke et. al. 1989), (b) Chaotic model of cognition.

A model for cognition involving chaos fig 18(b) applies similar principles. Incoming sensory information is coded in the form of a variety of structural transforms of the input representing aspects such as line-orientation, colour, tone, through to abstractions such as the 'grandmother' concept. These would arise as dynamical stability structures. Those aspects of a given problem which are consistent with the input constraints would attain coherence in their modules, extending the form of the input constraints leaving a complementary chaotic domain with a fractal boundary. The evolution of this system into a stable global regime, constitutes the solution to the problem. If however the dynamic retains chaotic islands which cannot be resolved from the boundary constraints, the problem remains incompletely solved, possibly requiring major reconstruction of the attractors and their surrounding stable regions - i.e. reformulation of the approach.

**(6) The Fractal
Extension of Chaos to the Quantum Level**

The discovery that fractal and chaotic dynamics plays a significant role in processes from the neurosystems level, down through the cellular and synaptic levels to the ion channel at the molecular level raises important new issues concerning the relationship between sensitive dependence and causal descriptions of neuronal dynamics. It becomes natural to ask the question "At what level does the fractal dynamics actually stop subdividing?".

In particular, since mutual interdependence can exist between unstable neurons and global neurosystems instabilities, the possibility of fluctuations at the molecular level, the quantal level of vesicles, or in the membrane being linked to global changes requires further examination. Usually nervous systems are believed to be subject to the laws of mass action and are treated as causal mechanisms which are not subject to quantum or other random fluctuations. Part of the reason for this is to ensure computing networks are robust to the kind of local damage which is seen to occur physiologically during the life of a nervous system. In particular, during the life of a human 100,000 cells may be lost a day, accumulating during a lifetime to around 10% of all cells in the central nervous system. Obviously brain function must have significant stability built in, which makes the system robust to such vagaries.

Nevertheless the extension of the notion of robustness to all aspects of central nervous function may be a serious conceptual error, which fails to take into account the need for sensitive dependence and structural instability, to guarantee tuning of the system to arbitarily subtle external stimuli, and to permit the development of new internal models, not provided by the structure of the circuitry at a given point in time as stability structures. Sensitive dependence of chaotic neurosystem dynamics may thus provide a mechanism in which fluctuations at the cellular, synaptic and ultimately the molecular level may become linked, so that fluctuations resulting from quantum uncertainty could become amplified to the global level. Mandel (1986) illustrates a similar hierarchical route in discussing the effects of lithium.

The nature of instabilities in chaotic systems requires only a small sub-population of cells to be in an unstable state. Such instabilities can easily be restricted to short-term dynamical characteristics which preserve long-term robustness. A holographic transform-based memory is naturally robust to partial loss of information, because cell loss or disturbance results only in a differential loss of discrimination of all stored items, and thus does not require microsystem robustness. Thus the possibility remains open that fluctuations at the cellular or even molecular level may play a part in resolving global instabilities through sensitive dependence.

We will examine these possibilities on each level of organization :

(a) Neurosystems: Sensitivity to initial conditions makes it possible for one or a few critical neurons to alter the stability of a much larger neurosystem of stable cells. Given a structurally unstable neurosystem which is about to undergo bifurcation, only a very small sub-population of critical cells need be in an unstable state to determine the stability-breaking of the global dynamic. However the crossing of threshold of such a critical neuron could conversely develop in a time-dependent cascade into a global bifurcation of a whole net. In principle this would make the neurosystems level sensitive to fluctuations at the level of a single neuron.

(b) The Neuron: A neuron whose threshold is tuned to its input level acts as an unstable bifurcator of a fluctuating input state. The occurrence of chaotic dynamics in single neurons and the use of mutual inhibition and threshold tuning in biological and neurosystems architecture provide a further basis for instability through sensitive dependence. The occurrence, for example in the hippocampus, of neurons receiving simultaneous input via distinct neurotransmitters from a variety of sense modes the reticular activating system and other sources supports mutual functional mapping between global neuro-dynamics and that of single neurons, which would enable single cell or micro-system instabilities to become global. The combination of unstable bifurcator cells and those with chaotic dynamics would provide a powerful means for amplification of such neuronal instabilities.

(c) The Synapse: The synapse is capable of both discrete and continuous transformation of the input into output. In cortical synapses, there is no need for the large number of vesicles in the neuro-muscular junction and it has been proposed that in some synapses, the release of contents of a single vesicle is sufficient to traverse the threshold and elicit a post-synaptic response. Even in the standard model the release of the acetyl-choline in a single vesicle, causes discrete micro-potentials which depolarize the membrane by about 1 mV, sufficient to result in an action potential if a cell is already at threshold.

(d) The vesicle: The vesicular structure in a sense results in the amplification of quantal instabilities from the level of the molecule to the larger level of the vesicle through the agency of the topological closure of membrane structure. The nature of vesicle exocytosis, involving about 10,000 neurotransmitter molecules and 2000 ion channels, generally appears to create a large enough population to produce causal mass action at the neuro-transmitter level. However the kinetics of vesicle association with the pre-synaptic membrane is determined by binding to one, or a few proteins, making vesicle release a function of the kinetics of one or a few molecules. The precise mechanism of vesicle exocytosis is not yet elucidated, but may involve binding to the membrane protein synapsin I. A quantum-kinetic model would be particularly relevant in excitation requiring only one or a few vesicles. Eddington (1935) and Eccles (1970) have raised the possibility of quantum-mechanical action of the vesicle and noted that the uncertainty of position of a vesicle of 400 oA diameter and mass 3 x 10-17g is about 30oA, comparable with the thickness of the membrane, making neurotransmitter release potentially subject to quantum uncertainty.

(d) The ion-channel: Activation of a single ion channel requires one or two neurotransmitter molecules. While the ion flux resulting from a single open ion channel will not generally elicit an action potential, if the channel happens to command a critical site on the two-dimensional dendritic surface, for example close to the cell body where the action potential begins, and the cell is at or near threshold, then the single quantal encounter of a neurotransmitter binding to an ion channel could evoke an action potential.

Sensitive dependence and quantum amplification could thus act together to make the brain potentially able to detect fluctuation at the quantum level. This is consistent with the sensitivity of sensory apparati which are all capable of detections at or close to the level of single quanta. For example the minimum number of photons required to elicit a response in a nocturnal animal, or the number of pheromone molecules required to elicit behavioral reaction in moths, are both close to unity.

The possibility of a connection between quantum mechanics and brain function has been a source of interest since the discovery of the uncertainty principle, partly because of its implications for consciousness & free-will. The connection between the observer's mind and quantum mechanics is pivotal in some interpretations of wave function collapse. Bohm's work on the Einstein-Podolsky-Rosen conjecture, Bell's theorem and the Aspect experiments (Clauser & Shimony 1978, Aspect 1982) which display spin-correlations between a split photon pair over space-like intervals have demonstrated that hidden variable theories must be non-local, leading several researchers to postulate the idea of non-local states correlating the activity of various parts of the brain (Penrose 1987). Collapse of the wave function appears to be the aspect of quantum mechanics which best supports the sensitivity and indeterminacy of chaotic systems, since the evolution of the quantum Hamiltonian appears to avoid true chaos. Popper and Eccles (1977) and Margenau (1984) have also discussed the possibility of quantum reduction being associated with the nature of free-will and the mind. Basar (1983b) has suggested matrix theory and Feynman diagram approaches (Stowell et. al. 1989) to resonances at the neurosystems level by drawing from ideas in physics in brain modelling.

However it is one thing to suggest that quantum fluctuations could in principle evoke global bifurcations of brain function, but quite another to determine what advantage might accrue from such seemingly stochastic activity. We will thus examine models which attempt to explain possible advantages of quantum uncertainty in brain function.

Deutch (1985) has analysed the potentialities of a quantum computer which has a 'fuzzy' logic representing quantum superpositions of states to form a probability function in the interval [0,1] in place of the usual {0,1} = {T,F} of formal logic. Although the algorithmic capacity of such a quantum computer does not extend the class of functions computable by a conventional Turing machine, several specific instances have been given in which a quantum computer might solve special tasks more efficiently, (Lockwood 1989). However these do not appear to provide significant advantages over parallel distributed processing. Both these authors adhere to the Everett many-worlds interpretation of quantum mechanics in which the collapse of the wave function never occurs, and all histories having a non-zero probability under the quantum prediction are presumed to co-exist as parallel aspects of a cosmic wave function. This places a specific limit on their capacity to utilize wave function collapse, and restricts their models to the use of superpositions of quantum states in place of a single physical state.

The mathematician Roger Penrose (1986,1989) has also studied the relation between the conscious brain and quantum physics in depth and attempted to combine quantum theoretic and relativistic ideas. He has suggested that collapse of the wave function may be a deterministic process based on the interaction of the superimposed wave function with the gravitational field at the level of one graviton. Gravity is the most difficult force to integrate into a quantum approach. The development of superstring theories and higher dimensional Kaluza-Klein space-times may establish a consistent theory of quantum gravity, making this suggestion open to more rigorous evaluation.

**For a full discussion of the dual-time model see also:**

**Fractal Neurodyamics and Quantum Chaos****:**

Resolving the Mind-Brain Paradox Through Novel Biophysics (1996)**Quantum Mechanics, Chaos and the Conscious Brain****(1997)**

The **dual-time model** (King 1989), which combines quantum
theory and special relativity, develops a supercausal theory which
is consistent with conventional quantum mechanics, but allows
for correlations between reduction events over both space and
time.

This replaces the stochasticity of the quantum model with a time-symmetric description, based on the transactional interpretation of quantum mechanics in which an exchanged particle is treated as a transaction between emitter and absorber. Because the boundary conditions of a transactional exchange necessarily include future states of a system, the initial conditions remain insufficient to fully determine the system implying that future states of a system influence reduction events by forming part of the boundary conditions. In systems in which only one or a few reductions take place for each state, convergence to the quantum probability interpretation does not occur. Such conditions arise when fluctuations lead to a sequence of new states, as is the case when they result in successive bifurcations of the dynamic. A system exchanging a restricted set of particles internally or externally might then gain access to a form of predictivity which could explain the evolutionary emergence of consciousness and free will as consequences of such causally indeterminate aspects of brain function.

The model thus proposes that what may appear to be quantum noise to an external observer may enable a form of predictivity to be utilized by transform processes in the generation of an internal space-time model. It should be noted the the global architecture of the pre-frontal cortex and limbic system (Alkon 1989, Mishkin & Appenzeller 1988) could be closely described as a holographic representation of time in which future and past states are both represented using transform features similar to those of the auditory and visual sensory processing. While memory represents past states, the orientation of organized goal-seeking behavior implicitly involves future states. Prefrontal lobotomies in both man and monkeys are known to effect both these processes (Alkon 1989).

Conscious awareness would in this perspective be identifiable with the quantum indeterminacy of major chaotic resonances involving the cortex, mid-brain and limbic system as exemplified in the EEG. It is also possible that the subjective sensory aspects of mind, including the qualitative differences between the senses as exemplified by colour, sound, odour and touch arise from the different ways these modes of stimulation act at the quantum level, as photon, phonon, orbital interaction and soliton rather than from a merely structural difference in feature detectors. Our 'internal model' may thus act by re-evoking such quantum transformations in the brain.

The development of chaotic excitations may have arisen in the earliest cells as a form of extra-cellular sense organ utilizing sensitive dependence. Electrochemistry has been proposed as a predecessor to genetic translation which may have emerged spontaneously as a stability structure in the 'RNA-era' (King 1978, 1990) before the development of coded enzymes, as a result of non-enzymic photo-electric ion transport. The ancient origin of molecules such as the amine neurotransmitters, and the components of phospholipids is notable in this respect. It is thus possible that electrochemical amplification of quantum chaos is a physical property fundamental to all eucaryote cells, providing a unique window on fundamental properties of quantum physics, which could be of cosmological generality.

Summary : The chaotic aspects of brain structure and dynamics have been discussed. The relation of chaos to fractal processes in the brain from the neurosystems level down to the molecule has been explored. It is found that chaos appears to play an integral, though not necessarily exclusive role in function at all levels of organization from the neurosystems to the molecular and quantum levels. An interesting consequence involving the possible interface between chaotic dynamics and quantum physics has been discussed because of its potential significance in resolving several of the most intractable conceptual problems to do with computability, the brain and the mind (Blakemore & Greenfield 1987, Hooper & Teresi 1987, Rose 1973, Searle 1979, Penrose 1986, 1989).

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NOTE: All diagrams are digitally processed by the author. All original sources are indicated in the text or captions.