The structure of chaotic attractors in perception
The combination of continual and unpredictable local variation with long term constancy is shown by the two sets of EEG records in Figure 12 lasting 5 seconds and taken 15 seconds apart. The upper trace (A) is from the olfactory bulb, B is from the anterior nucleus, and C-E are from the olfactory cortex. The bursts occur with inhalations that bring the system out of its low amplitude basal chaotic state. The successive traces are more similar to each other than the traces from the different parts, and the three traces from the cortex are more similar to each other than to those from the other parts. The constancy is such that an experienced observer can tell at a glance the source of a trace, but the variability is such that no prediction can be made as to the detailed future time course of any of the traces.
The same combination of variability and constancy is found in the spatial AM patterns. Each pattern is described by the 64 amplitudes of the common wave form and therefore by a 64x1 column vector or a point in 64-space. For display purposes a set of points from a number of bursts can be projected into 2-space while preserving the relative distances between them (Sammon, 1969). The variation in pattern is shown by the dispersions of the clusters from two animal subjects, and the constancy is shown by the clear separation of the two subjects (Figure 13, upper frame). Each animal has its own unique pattern, but the expression of it varies randomly from each burst to the next in the absence of deliberate odorant stimulation, as shown by the nearly symmetric projection of the points for the burst from each animal alone (Figure 13, lower frame). In Figure 14 the dimensions have been reduced to 2 by means of step-wise discriminant analysis (Freeman and Grajski, 1987) for a data set from one subject containing bursts in 3 conditions: control, a conditioned stimulus odorant paired with a reward (CS+) and an unreinforced odorant (CS-). The centroids of the three groups are denoted by the large (0, +, -). The inset table shows the classification of the individual spatial AM patterns with respect to the centroids by a Euclidean distance measure. The variability in pattern is shown by the spread of the individual points, and the constancy is reflected in the grouping around the centroids. The rate of correct classification (72%) is typical for these data and is only slightly below the rate of correct performance of the conditioned response by the animal.
Whenever a new odorant is introduced and a new pattern forms for that odorant, all of the other patterns change as well. Global AM pattern changes accompany the switching of reinforcement contingencies (CS+ and CS- reversal), even though the odorants and the responses are the same. When an old CS+ is re-introduced, its spatial AM pattern changes to a new form, not the old one (Figure 9). These findings lead to conclusions that the macroscopic AM patterns are not invariant with respect to stimuli, and that they reflect instead the meaning of the stimuli for the subject. Perhaps most significantly, the patterns cannot be derived from the stimuli by filtering. They are endogenously created by the bulb after its synaptic connections have been modified during learning.
One interpretation of these data is that the neurodynamic machinery of the olfactory system develops and maintains a global chaotic attractor, which resembles a Lorenz attractor in having wings, but instead of two it has multiple wings, one for each class of odorant that it can discriminate. In this view the system when left without input settles into a basal chaotic activity that can be maintained for indefinitely long time periods. When receptors are stimulated by a learned odorant, the system is forced out of the basin of the interburst attractor by the amplitude-dependent nonlinear gain (Figure 11) of the populations (Freeman, 1979) and is constrained in one wing of the global attractor (Freeman, 1992).
The system does not learn single episodes. It learns to assign inputs into classes. This property reflects the nature of the operation that the olfactory system performs, which is to generalize over its specific inputs. There is a very large number of receptor neurons in the nose, on the order of 108, and about 105 of these may be capable of responding to any one odorant. On any one inhalation only hundreds or perhaps thousands of receptors actually receive the odorant, but because of turbulence in the nasal passage, the selection of the 103 or 104 from the 105 receptors available is different on every trial. A nerve cell assembly can form during training by strengthening synapses among the olfactory bulb excitatory cells that are co-excited by input, and it thereafter responds in a stereotypic way when any of its members are excited in any combination. The system is forced out of the basal state by receptor input with a known odorant and is constrained to oscillate in one of the wings of the global attractor. For this reason the oscillation can be visualized for only one wing with each inhalation and then only briefly (Figures 3, 12).
The geometry in state space of EEG attractors can be visualized by plotting two or more traces of maintained activity against each other or by plotting one trace against itself lagged twice in time as shown by example (Figure 15). An analogy is to plot on a map of a city those routes that are taken by a salesperson making routine rounds to customers, and from home to office and back again. After several weeks certain preferred trajectories become apparent, in which there is a prominent home-office loop and a loop for each preferred customer. The collection of loops forms a global attractor, and the appearance of a certain loop on any given day discloses an input from a preferred customer. The introduction of a new loop (customer) corresponds to a bifurcation with learning a new odor in olfactory dynamics, instituting a structural change in the global system under associative learning.
This analogy breaks down for brain dynamics, because there is no equivalent of a city map, and olfactory dynamic space not restricted to 2 dimensions. At least 3 and preferably 4 dimensions can be used for display of EEG phase portraits with computer graphics resembling a flight simulator to manipulate the structures in 3-D with color to display EEG amplitudes in a fourth dimension. Extended raw EEG records give phase portraits that are too complicated to reveal meaningful structure. There are several reasons. One is that the bursts and interbursts are too brief to trace out the forms of chaotic attractors. Successive bursts trace different parts of the wing for the same odorant and different wings for different odorants. Other parts of the brain participate with the olfactory system during perception of an odor, which adds to the complexity of the trajectory. Yet another is that raw EEGs have contributions from multiple sources, many being still undefined, that perturb the trajectories and distort the phase portraits. Finally, the defining properties of a wing are its spatial AM pattern and its basin of attraction with respect to the domain of input, and these properties are not readily apparent in displays of 64 simultaneously recorded time series.
In summary, the olfactory system comprising the bulb, nucleus and cortex maintains a global attractor with many wings observed one at a time can be observed. A wing of the attractor in state space looks like a loose coil of wire. By hypothesis an act of perception consists in the explosive leap of the dynamic system from a chaotic basal state to a wing or from one wing to another under the selection of a nerve cell assembly by a stimulus. An act of associative learning is a bifurcation, an irreversible structural change in the global attractor that adds a new wing and modifies existing wings.