Walter J. Freeman Journal Article e-Reprint

Qualitative Overview of Population Neurodynamics

Neural modeling and neural networks, 1994, 185-215

WALTER J. FREEMAN

     Much of what is understood about the functions of these massive numbers of neurons comes from recording the activity of neurons one at a time with microelectrodes. Networks of model neurons are constructed by induction that represent the actions of a small subset of neurons, on the implicit assumption that what the other (unobserved) neurons are doing during a perceptual act or a conditioned reflex is not important for the activity of the observed subset. This approach might be called statistical mechanics (Wilson and Cowan, 1972; Amari, 1974; Amari and Maginu, 1988; Hansel and Sompolinsky, 1990) of brain function, comparable to the study of molecules in an ideal gas in terms of their kinetic energy, position, collision rates, etc.

     A different approach is to conceive not individual neurons but interactive populations of neurons. The weak interactions among immeasurable numbers can be predicted to give rise to cooperative properties, which can only be accessed by measuring local mean field quantifies constituting observable macroscopic variables for the masses. This alternative approach might be called a thermodynamics of brain function, by which the equivalents are sought for the temperature, pressure, viscosity, etc. of an ideal gas. However, physical analogs on their face are inappropriate and have led many physicists astray. Neurons are not molecules in a gas, so a fresh beginning must be made for a new brain science (Skarda and Freeman, 1987).

     Neural populations come into play significantly during animal behavior and their properties cannot be understood from studies of isolated neural preparations in vitro or in brains under anesthesia. Therefore this new approach is introduced in the context of acts of animal perception and in particular the perception of odors through the olfactory system (Fig. 1), because it is the simplest, and best known at present and because there is good reason to propose (Herrick, 1948) that it is the phylogenetic prototype for the algorithms by which all sensory systems operate in perception and indeed for all instances of behaviorally related interactions of cortex within itself and with subcortical nuclei and the far reaches of the brainstem and spinal cord.

     Consideration is restricted here to acts of pre-attentive perception that do not require inspection or combining successive stimuli over time. Such acts occur in each sensory system independently of other senses on first arrival, but in the time to the next heartbeat, eye movement, or inhalation, all the senses are involved. This takes place by convergence from several sensory cortices into the limbic and motor systems of the brain (Fig. 1). These acts of perception occur in all vertebrates and perhaps in all animals. The acts are rapid in onset, short in duration, reliable and reproducible. In animals they are easily controlled by standard techniques for conditioning. For these reasons a rabbit that has been trained to sniff an odorant is an optimal subject for the study of the neurobiology of perception in the olfactory system, with the hope that what is learned about perceptual coding will hold for all other senses as well (Freeman and van Dijk, 1987; Tsuda, 1991).

     It is known from lesion studies, in which the damage is caused by disease or experimental surgery, that the neural activities sustaining these acts of pre-attentive perception take place in the outer shell of the brain, the cerebral cortex. From behavioral experiments with trained animals it is known that each act takes place just after some sensory input has been transmitted to each of the sensory cortices in the brain and that it then takes place within a few tenths of a second of stimulus arrival, before any decision is taken on what to do next. Response latency measurements show that it happens in the time needed for a sniff or a saccade ("a twinkling of the eye"). These findings show exactly where and when to look for the behaviorally related physiological process in the brain that sustains a percept.

     Previous studies in neuronal physiology (e.g. Shepherd, 1983) have shown how sensory stimuli are analyzed, not how percepts are synthesized in the brain. From studies combining theory and experiment it is known how sensory stimuli are transduced by receptor neurons in a two-stage process. First the stimulus is converted to a loop current with its energy source in the membrane at or near the site of action of the stimulus and its effective action at the initial segment of the sensory axon, where its amplitude is re-expressed by the frequency of a train of action potentials. The loop current flows inside the cell in one direction and outside the cell in the other direction across the tissue resistance, giving rise to a "generator potential", which for the olfactory receptors in the nose is called (Fig. 2) the electro-olfactogram. This two-stage operation also serves as a model for the function of cortical neurons.

     They receive action potentials ("units") at the synapses on their dendrites and convert them to dendritic loop currents. The currents are summed at the initial segments of their axons (the "trigger zones") where the net amplitude determines the frequency of axonal firing. The dendritic currents cause oscillating waves of electrical potentials to appear in and around cortex (Baser, 1980) called the electroencephalogram (EEG). Thus there are two main state variables by which neural activity is carried: pulses on axons and waves on dendrites.

     Similar pre-analysis also takes place in the olfactory system, but the processing is far less complicated and largely consists of dynamic range compression and signal normalization (Freeman, 1975), though some degree of contrast enhancement may take place (Shepherd, 1983). The simplicity makes easier the task of finding percepts. The olfactory system has a rough topographic map from its receptor neurons in the nose into the receiving sheet of cortical tissue, the olfactory bulb (Fig. 3).

     Roughly 50 million olfactory receptors in the rabbit transmit their out by unbranched axons to half a million neurons in the bulb. As is typical sensory systems a large number of receptors converges to a small number of bulbar neurons, the convergence ratio here being about a thousand to one.

     The axons of the bulbar neurons that carry pulses to the olfactory co] do not form a topographic map. Each output axon has many branches that diverge widely over the cortex. Conversely, each cortical neuron receives input from neurons that are widely dispersed throughout the bulb, thereby performing not only temporal integration but spatial integration as well. This divergence is an important property which will repeatedly referred to in this review. This type of divergent connection may in fact be much more common in cortical connections than is the topographic map.

     The studies in olfaction will now be used to answer the question of how, within a few tenths of a second, an act of pre-attentive perception is accomplished, that is, how relevant sensory input is (a) extracted from the environment by a sniff, (b) globally integrated by arrays of cortical neurons and (c) combined with pertinent past experience and expectancy of future events into a space-time pattern of cortical activity. That pattern is called a "wave packet" in its physical aspect and a "percept" in its behavioral or functional aspect (Freeman, 1975). The answer will be expressed in the language of nonlinear dynamics of neural populations.

     The basic anatomy and physiology of neurons is familiar in terms of the Neuron Doctrine, which holds that the element of neural function is the individual neuron. Complementary to this doctrine is the concept of the cooperative neural mass, that exists through synaptic connections among participating neurons (Fig. 3). In this hierarchical view the population activity co-exists with the trains of action potentials from individual neurons in the form of shared spatially distributed patterns of activity occupying large areas of cortex, having sudden onsets and offsets and lasting small fractions of a second. These wave packets have only recently become available for measurement with large arrays of electrodes and amplifiers. They form the foundation of this new approach.

     The concept of coexisting "microscopic" and "macroscopic" activity is familiar. Each person as an individual speaks with one voice, but each participates in families, committees and economic units. Those larger participations become apparent only in the aggregate, such as committee reports, vote tallies or economic indices. Similarly, each animal is also part of a species, a food chain and an ecological system. A water molecule is simultaneously part of a hailstone, a cloud and a weather front. The challenge of applying this concept to brain function is not theoretical. It is experimental. How can one observe the activity of a neural mass? How can one determine that what one sees relates to behavioral brain function? The collective activity must be observed as a sum or an average of the activity of members of the population. Like a census taker one must collect data from individuals, make averages over samples and then find the behavioral correlates of the results, which have no meaning unless and until they are related to what the brain is doing.

     One must choose between two methods of averaging. A single amplifier suffices for recording a trace of activity from one neuron in a mass or from a local group of neurons acting in concert (Fig. 2). The recorded trace typically lasts merely a few seconds or even a fraction of a second. On repeated stimulation one can collect and store an ensemble of traces over time. If one aligns these stored traces using the stimulus time marker, one can compute an average trace called an ensemble time average, also known as an averaged evoked potential or event related response (Baser, 1980). With many electrodes in a spatial matrix one can simultaneously obtain an ensemble of traces each time the stimulus is presented (Fig. 4).

     However, by using the stimulus time marker to align successive evoked potentials from serial stimuli, one can also compute a spatial ensemble average for the single stimulus presentation on every channel (Fig. 5). In this example it is computed from the collection of averaged evoked potentials instead of from the unaveraged time traces.

     Profound differences exist between these two averages. For example, suppose that a chorus sings a musical phrase under the baton of a conductor. The listener hears a spatial ensemble average from the singers, who also hear and respond to each other interactively. Now suppose that the conductor were to ask each member to sing the phrase solo while the sound is taped, using a tap of the baton to start each singer. The conductor could make a time ensemble average by aligning the tape segments at the tap and summing the sounds to reconstitute the chorale. The reason this method cannot give the same result as simultaneous singing is that the singers do not have the necessary precision of pitch and timing unless they can interact to form a population as they sing.

     Time ensemble averaging works well for neuronal events that are solidly time-locked to stimuli, but it does not work on population events, because each collective cortical event is internally generated. The onset time of each event is related but not bound to the stimulus and variable in latency. Endogenous events typically consist of brief fluctuations that vary in their frequency both within particular events and over successive events (Fig. 2). These variations cause destructive time ensemble averaging. A spatial ensemble average of a collection of traces that are recorded simultaneously, in contrast, tends to wash out local detail but to emphasize the common wave form of the population.

     Time ensemble averaging can be used to display the impulse response of cortex to its sensory input and spatial ensemble averaging can be used to derive an estimate of its perceptual output. This is shown by comparing two forms of electrical field potentials recorded from the olfactory bulb. First, an electrical stimulus delivered to the afferent pathway leads to a characteristic damped oscillation. This can be recorded with an array of 64 electrodes. When the stimulus is low intensity, the response of the bulb is an oscillation at 40-60 Hz like the ringing of a bell when it is struck (Fig. 4). The response is restricted to a part of the bulb to which the stimulated axons carry the input. It has the same frequency of oscillation everywhere over its duration and its amplitude pattern reflects accurately the spatial pattern of the input (Freeman, 1991). In other words, this is a stimulus bound response of the bulb. Because the oscillation occurs only when the input intensity is low, it is obscured by the ongoing EEG, so it must be extracted by repeated stimulation and time ensemble averaging at each of the 64 channels. Measurement of the spatial pattern of phase of the 64 damped cosines reveals a time lag of the oscillation corresponding to the direction and speed of propagation of the afferent axons that carry the electrically evoked action potentials (Fig. 5). The contributions of the EEG are removed by this approach, along with the amplitude and phase patterns of the endogenous bursts, because the times of onset of the bursts and their frequencies of oscillation vary unpredictably from one burst to the next without precise relation to the times of onset of the electrical stimulus.

     In contrast, the normal input is from olfactory receptors that are activated by inhalation, which induces a burst of oscillation in the EEG (Fig. 2). The burst involves the entire bulb (Bressler, 1984). It is observed through the same array of 64 electrodes placed on the bulbar surface, serving as a 4 x 4 mm window. A typical burst with one inhalation has a duration of about 100 ms with 4-6 cycles of oscillation at typical frequencies of 40-60 Hz. There is a common wave form distributed over the entire bulb, which is derived by statistical analysis and it is found to have a spatial pattern of amplitude modulation (AM) (Fig. 6). Furthermore, perceptual information is carried in this spatial AM pattern.

     This burst is not evoked by the input. It is induced, because the input makes the bulbar populations unstable. The instability is manifested when the bulb jumps to a new state and creates a new spatial AM pattern of activity. The evidence for this state change is found in the spatial pattern of the phase of the common oscillation. This has the form of a cone, which implies that each transition starts at one point, called a site of nucleation and spreads uniformly in all directions at rates near 2 m/s, the velocity of axon collaterals in the bulb (Freeman and Baird, 1987). The location of the apex of the cone is a random variable between successive bursts and can be found anywhere on the bulbar surface unrelated to the odorant stimuli that select the spatial AM pattern (Fig. 7).

     The polarity of the apex (maximal phase lead or lag) is also a random variable, so that the point cannot serve to "locate" a "pacemaker" neuron. Hence the phase pattern proves that the bursts are endogenous or "self-organized".

     There is a yet more profound aspect of spatial ensemble averaging to be considered. Spatial integration is the mechanism by which neurons in a local neighborhood corresponding to a cortical "column" make their own spatial ensemble average and thereby establish the basis for cooperative collective action. Spatial integration occurs continually in each neuron as its dendrites converge and sum the waves of current that are generated by its synapses upon axonal pulse input. It also occurs as each neuron transmits to many others in the surround and then receives their transmissions back again. In many if not all areas of cortex the excitatory neurons excite each other (but not themselves) and the cooperative interaction is the basis for the emergence of the local mean field and the population neighborhood. Receptors cannot form a population in this sense because they are not synaptically connected and they are not globally interactive in other ways. Spatial integration also occurs between cortical areas, as for example in the divergence of the axon projection from the bulb to the olfactory cortex, which is important for the readout of the population, but divergence alone and parallel action alone are not sufficient. There must be feedback, as indicated by feedback within the transmitting population and by the "centrifugal" connections (Fig. 1) from the target populations back to the transmitter population.

     The macroscopic activity of the population is difficult to observe in the pulse activity of the individual neurons, because a large amount of random variation exists in the intervals between pulses in the "spontaneous" background activity of each cell. From some indirect measures it is estimated that in the cortex only one part in 1000 to 10,000 of the total variance in the activity of the individual neurons in the population is covariant with the activity of the population. Because time ensemble averaging destroys the endogenous components, the minimum for detecting population activity would require simultaneous recording from 1000 to 10,000 cells. At present the experimental limit for recording "units" simultaneously is from about 100 individual cells.

     Nature, however, provides easy access to cortical spatial ensemble averages. This comes about because the same dendritic currents that determine the firing rates of single neurons at trigger zones flow across the tissue resistance and their ohmic potentials add as the currents pass through the tissue (Fig. 8). When the extracellular field potential is properly recorded, it provides a basis to estimate the strength of the local mean field activity in the neighborhood of the neuronal population from the amplitude of its EEG.

     Multiple simultaneously recorded EEGs from an array of electrodes placed on the cortical surface at suitable spacings provide a basis for observing the spatial AM patterns of cortical activity (Fig. 6). Other methods currently under development include measurement of cortical electric fields with voltage-sensitive dyes and optical recorders, (T'so et al., 1990) or with magnetic sensors to measure the magnetic components of the fields of dendritic current (Williamson and Kaufmann, 1989; Llinas et al., 1991). At present their instrumental noise levels are so high that they cannot be used without time ensemble averaging or narrow band pass filtering, so the use of these methods is restricted to measuring time ensemble averaged evoked potentials and not the unaveraged traces.

     Simultaneous recordings of EEGs from 60 to 64 electrodes in rectangular arrays placed surgically on the olfactory bulb or cortex reveal a substantial degree of spatial coherence of the activity at all times. In contrast, the time series recorded at each electrode is highly irregular. The wave form from each electrode is as nonreproducible as a freehand scrawl, but the simultaneous recordings always contain the same scrawl (Fig. 6). To be sure, there are local differences in the amplitudes and peak latencies (Fig. 7) of the scrawl, but the same peaks and troughs occur everywhere and the instantaneous frequency of the EEG traces tends to be the same.

     The commonality is often difficult to see in the visual, auditory and olfactory cortices, because the EEG reflects the sum of electric currents from multiple sources, many of them unknown and the common wave form comprises only half to three quarters of the total variance of the activity in the traces (Freeman and Viana di Prisco, 1986; Freeman and Grajski, 1987). The common wave form is extracted by computing a spatial ensemble average from each set of EEG traces, in much the same way that the cortex extracts the bulbar output. This commonality of wave form extends over the entire bulb. Similarly other common wave forms cover the entire olfactory cortex (Bressler, 1987) and much if not all of the primary visual (Freeman and van Dijk, 1987) and auditory (Pantev et al., 1991) cortices. In contrast to lower frequencies such as the (8-12 Hz) and ø (3-7 Hz) rhythms these time series appear to be self-organized within the cortices and not imposed by pacemakers lying in the basal ganglia or brainstem.

     What makes these spatial AM patterns interesting is the fact that they contain behavioral information as demonstrated experimentally by conditioning. After an animal is trained to discriminate and respond to an odorant (Fig. 9), a unique spatial AM pattern reappears in the EEG of the bulb whenever the animal inhales that odorant.

     If it inhales another odorant that it can discriminate, a different reproducible pattern emerges. The common oscillatory activity constitutes a "carrier" wave and the information about an odorant is expressed by the spatial AM pattern of the carrier in the wave packet independently of its wave forms, frequencies, or latency patterns.

     These findings suggest that olfactory coding resembles the frames in a movie film. The light is a fluctuating energy or carrier (wave form) that bursts on and off with each frame. The information in each frame (sniff) is given by the intensity of the light (amplitude of the wave) at each pixel (local neighborhood) and it is held for the duration of the frame. If the same scene is shown in sequential frames, the spatial pattern of the carrier amplitude is reproduced in each frame. If a new scene appears, a new spatial pattern is seen. This analogy may suggest the intermittent flow of perceptual information in the olfactory system by a process known as chaotic itinerancy (Tsuda, 1991).

     To recapitulate, microscopic coding is found in its clearest form in the peripheral nervous system and in the sensory and motor relays to and from the cerebral cortex. The information is carried in the pulse trains of a selected subset of neurons. Observers extract its statistical time series analysis of point processes using time ensemble averaging on the recorded data, because each pulse sharply defines the time and place of its piece of information in terms of the time interval from the last preceding pulse at that trigger zone. The pulse frequency gives the intensity of the stimulus and the origin of the axon determines the quality of the stimulus. Stimulus parameters are modified by cellular mechanisms that control receptor sensitivity, but they are not changed by associative learning. Macroscopic coding is found most clearly in cerebral cortex, co-existing with the microscopic activity that is injected into the cortex by afferent axons. This information is spatially distributed over the entire population comprising an area of cortex, but it is resolved into time segments. Within those segments it is stationary in the form of an amplitude modulation of a common carrier. The patterns are self-organized, not topographically related to input and strongly dependent on modifications by near and remote experience that has been incorporated into the cortex by learning.

     Continuing with olfaction, the EEG recordings reveal changes in the spatial AM patterns of the carrier wave. The changes imply that synaptic strengths are modified with associative learning. How and where do the changes with reaming take place? The answer is found by training an animal to discriminate an electric stimulus given to an olfactory pathway and measuring the monosynaptically initiated impulse responses of the bulb and olfactory cortex (Freeman, 1975). The averaged evoked potentials (similar to those in Fig. 4) do not increase in amplitude but show greater durations in the first upward peak and in the subsequent oscillations. They are less stable in the oscillatory mode. This pattern can only occur when some of the synapses that interconnect excitatory neurons within the olfactory bulb and cortex are selectively strengthened (Fig. 3). The increase in synaptic strength with reaming does not, as is commonly supposed, take place at the synapses between the incoming axons and the excitatory neurons, because the initial amplitude does not increase. It appears instead that, in accordance with a variant of the Hebb rule, which holds that synapses between neurons that fire together are strengthened, the synapses that are augmented are only those between bulbar neurons that are simultaneously excited by the training stimulus. The synaptic changes with behavior only occur when the training stimulus is an odorant and when it is accompanied by a reinforcing stimulus that is rewarding or punishing. The data show that over a sequence of a few dozen sniffs on as few as four or five trials a "nerve cell assembly,' forms for a particular odorant as the animal reams to discriminate it. The assembly consists of a small subset of cells that were co-excited on one or more reaming trials on which reinforcement was given. This combination of conditions strengthens the synaptic connections between them (Gray et al., 1986). When any subset of neurons in this assembly thereafter receives input, the entire assembly is excited by the enhanced mutual excitation. These principles have been embodied in sets of differential equations, the solutions for which give spatial AM patterns of chaotic carrier wave forms after modification of connection strengths by reaming in accordance with a form of the Hebb rule (Freeman, 1987; Yao and Freeman, 1990).

     Studies of the EEGs from these animals and in particular the phase patterns (Fig. 7) show that with each onset and offset of a burst there is a dramatic change or state transition of the bulbar population, consisting of an abrupt change from one global activity pattern of the neural population to another one and from one set of properties to another set. The change is induced by the input and the new pattern is held or "clamped" until the input subsides.

     The concept of a global state transition in the brain is familiar from behavior. Both sleeping and waking are global brain states with abrupt transitions between them. During standing, walking and running the bones, muscles and neurons are the same, but their global patterns of periodic activity are susceptible to sudden changes. The analysis of EEGs is complicated by the fact that the carrier is irregular in its wave form and it never repeats itself, whereas those state transitions that are best understood are typically from one steady state to another, like water on a pond freezing or melting, or a muscle from relaxation to contraction. Another well known transition is from a steady state to rhythmic oscillation known as the Hopf bifurcation. This concept can be used to describe the change in an animal at rest that starts to walk or to swim.

     These dynamic modes form a well known hierarchy of stable states. The stability of a state of a system is determined by perturbing the system by brief input such as a pulse to determine whether it returns to the initial state. That state is characterized as an "attractor" owing to its perceived tendency to draw the system to itself and the set of inputs for which this tendency holds is called the "basin" of attraction, in analogy to the tendency of an object to move downhill to rest. The simplest attractor is a point, which represents zero change. This holds for single neurons at rest, for isolated slices of cortex and for brains under very deep anesthesia or in legally defined "brain death". The next is the limit cycle attractor characterized by a single frequency to which the system returns in periodic oscillation after perturbation. This holds for single neurons that are firing pulses at a constant frequency, for populations in rhythmic activity such as in locomotion and to an adequate approximation for cortex in respect to its capacity for generating regular EEG oscillations in the alpha and theta ranges. The more complex quasi-periodic attractor has two or more discrete frequencies, for which the spectrum consists of a spike at each frequency.

     The steady states that aptly characterize normal cortex differ from these three attractors in that the activity is not periodic. There is no one frequency or discrete number of frequencies into which the activity can be decomposed. The single neuron has a pulse train for which the interval histogram conforms to the Poisson distribution. The autocorrelation function has no sustained periodic oscillation (Fig. 10, upper frame).

     Yet the statistics of the pulse train are time-invariant whether or not the neuron has been transiently perturbed by external input and is observed after the transient (the evoked activity) has died out and the conditional probability of firing pulses on the local mean field of the population in which the neuron is embedded oscillates in the same manner as the EEG of the population (Fig. 10, lower frame). The EEG may often seem to be close to periodic, but its spectrum is broadly distributed about certain preferred peaks and its amplitude histogram is Gaussian (Freeman, 1975). Activity that is broad-spectrum and locally unpredictable at first glance might be characterized as "noise", but the population activity that is manifested in the EEG has spatiotemporal structure and global predictability. The structure can be observed in the common aperiodic carrier wave that is shared by many millions of neurons comprising an area of cortex, by the phase gradient that precludes the existence of a "pacemaker" for the common activity and by the ability of the cortex to return to a definite spatial pattern in circumstances of receiving an appropriate learned input. These properties clearly indicate that global cortical activity does not consist of unstructured noise, but it is a manifestation of deterministic chaos (Stewart and Thompson, 1986; Basar, 1990).

     The stability of a chaotic brain state is easily shown by use of perturbation. When an electrical stimulus is given that drives a sensory cortex away from its pre-stimulus basal state, it returns to that state by a characteristic trajectory that is revealed by an evoked potential (Fig. 4). The wave form of the evoked potential changes with the parameters of the stimulus (location and intensity, for example) but the state to which the system returns is the same by a variety of statistical assays for a specifiable domain of the parameter values. Therefore the basal state can be said to manifest an attractor and the domain of the input parameters specifies a basin of attraction for that attractor.

     More generally, the form of the activity of a sensory cortex as revealed by its EEG shows striking changes with modification of the behavioral state, as from sleeping to waking or from resting to aroused in hunger, fear, etc. Each identifiable state that can be shown to be stable under perturbation can be said to manifest an attractor for that cortex and the change in dynamics from one attractor to another can be described as a bifurcation. The change implies that at least one parameter within the system has changed and this is called a bifurcation parameter.

     An example of a bifurcation parameter is shown in Fig. 11 by the height of the sigmoid curve. Each neuron sums its dendritic currents and converts the instantaneous amplitude to a probability of firing. This probability is governed by the properties of the membrane, mainly the voltage-dependent sodium conductance, which determines the concave-upward part of the sigmoid and the voltage-dependent potassium conductance, which determines the convex-upward part of the curve. In a local population the output is a continuous pulse density as a function of the local mean field potential (Freeman, 1979; Eeckman and Freeman, 1991).

     The lower of the two sigmoid curves shows the slope and maximum for an animal at rest. The mean level of dendritic potential and of pulse density is shown by the small triangle. In this state there is basal chaotic activity, but no bursts occur. When the animal is aroused, the mean values, the slope and the upper asymptote increase with a single parameter. In this state the basal activity persists and a burst occurs with each inhalation, showing that the system has a different attractor. Bifurcation has occurred. When the animal is given a large dose of an anesthetic, the basal chaotic activity vanishes and the system bifurcates to yet another attractor, a point that is stable under perturbation.

     The burst is dependent on the presence of input during inhalation and not on a parameter change in the system, so it does not manifest a bifurcation. It is due to the asymmetry of the sigmoid curve, such that its steepest slope is not at the rest point for zero input (the triangle) but to the right (excitatory) side of the sigmoid. The two heavy curves show the derivatives of the sigmoids, which constitute the nonlinear gains. They show that excitatory input not only increases the output of a population. It also increases the sensitivity and the input-output gain. This feature depends upon the property of the individual neuron, that as it is brought closer to its firing threshold, its probability of firing increases exponentially. Automatically this increases the feedback gain between coupled populations, so that they tend to be destabilized. If the input has an odorant that activates a nerve cell assembly and if the sigmoid curve is sufficiently steep, then the population undergoes a state transition and breaks into an endogenous oscillation, a burst. The spatial AM pattern reflects the-previous experience embedded in the strengthened synaptic connections of the assembly. When learning takes place and a new assembly forms, the change in synaptic strengths constitutes a parametric modification of the system and a bifurcation can be said to have taken place, but accessing the new spatial AM pattern follows and does not constitute the bifurcation.

     The combination of continual and unpredictable local variation with long term constancy is shown by the two sets of EEG records in Fig. 12 lasting 5s and taken 15s 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 64 x 1 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 (Fig. 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 (Fig. 13, lower frame). In Fig. 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 reintroduced, its spatial AM pattern changes to a new form, not the old one (Fig. 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 reamed odorant, the system is forced out of the basin of the interburst attractor by the amplitude-dependent nonlinear gain (Fig. 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 (Figs 3 and 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 (Fig. 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 reaming 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.

     Given the synthesis of a wave packet corresponding to a percept, the next question is, how is it conveyed to those parts of the brain that will respond with it? The AM patterns are observed in dendritic potentials, but they are conveyed by action potentials of the many neurons from the bulb and widely disseminated over the olfactory cortex. While the commonality of the carrier of the population output is limited to a large fraction of the total macroscopic activity manifested in the EEGs, it consists of a very small fraction of the microscopic neuronal activity in the bulb, perhaps only one part in 1000 to 10,000. How in the next stage is the population output extracted by the receiving neurons under the conditions of an extremely low signal to noise ratio?

     One explanation is that because the output of each bulbar neuron diverges widely over the cortex (Fig. 3), each cortical neuron sums input from neurons scattered widely over the bulb, thereby effecting a spatial integration over the bulbar population. The only activity that survives this integration is that which has a common instantaneous frequency, provided further that the phase dispersion of the carrier is under a quarter of a cycle of the dominant frequency of oscillation. All other activity averages toward zero. But the common activity is the carrier. It is the cooperative output of the population that creates and transports the perceptual information and the measurements of the phase gradient of the carrier show that its spread is under the specified limit to avoid degradation by phase dispersion.

     In a word, divergence in the bulbar output pathway "launders" the bulbar message and it can only do so if the carrier oscillates (Freeman, 1991). The divergence will do this optimally if the undesired components are not spatially coherent, that is, at different frequencies at different points in the transmitting cortex. This answer explains the utility of the divergence in the output pathway of the bulb and also of the oscillatory carrier. However, this answer cannot account for the utility of the chaotic carrier, because a limit cycle carrier or a noise carrier would serve as well if it were spatially coherent. Then why is brain activity chaotic? This fundamental question remains unanswered, but there are some directions in which to search for answers. One suggestion is that chaos is inevitable in so large and complex a system as the simplest brain with its many interconnections. Engineers and computer scientists already find undesired chaos emergent in their more advanced systems. If this is the source and significance of brain chaos, perhaps the study of brain dynamics can show how to function in chaos.

     Yet the evidence suggests that brain chaos is the result of design by evolution and is not an epiphenomenon of hypercomplexity. Brains are intrinsically unstable. Even when animals are at rest and immobile their brains ceaselessly generate temporally unpatterned activity reflected in their EEGs. The chaotic olfactory EEG is not, as is commonly said, "a noise like the roar of a crowd at a ball game", because the EEG is not the smoothed and degraded sum of single cell activities. It is a deterministic output like a complex modern symphony that is orchestrated by the system. It is not a product of the bulb or cortex alone but of these and other parts of the olfactory system in feedback interaction. If they are surgically separated from each other, the activity vanishes (Freeman, 1975).

     One role for this mechanism might be to provide unpatterned activity as a form of exercise for neurons, which must keep active or die. In contrast to this view the "grandmother cell" hypothesis requires that neurons wait in silence through days or years for their trigger stimuli to appear and then fire faultlessly on feature presentation. This is biologically unrealistic. But a more important postulated role of chaos for brain function is predicated on the fact that chaotic systems not only destroy information; they create it. The essence of a perceptual act may consist of the neurodynamics which creates a neural activity pattern from the raw materials of stimulus, experience and expectancy. The greatest achievement by an act of perception is the immediate creation of the meaning for the subject of some relevant information from a continuous environmental inflow that is complex beyond all measure.

     Man-made information processing systems face this same infinite rate of information flow and they handle the task by use of filters (Fig. 16). The filters define what is "signal" and remove everything else as "noise".

     In all cases it is the engineer or observer who decides what is signal and what is noise, either by direct construction of the filter or by use of a reference standard with which to compute an error, so as to shape a filter through a "teaching" procedure as in back propagation. Such an agent in a brain would be an homunculus and is not admissible. Brains do not allow direct intervention by outside observers to determine what is feature or signal and what is to be ground or noise. Brains act to organize themselves and to shape the environment and the relations of their sensors to the environment, as in sniffing or moving the eyes. The raw act of perception by hypothesis may be replacement of an infinite environmental inflow with a finite dimensioned chaotic activity pattern. The fractal dimension may be determined within the system and may vary according to prevailing circumstances and constraints. The requisite perceptual meaning may be freshly minted under the stimulus conditions and the state of the brain holding at the instant of the state transition of the system away from the basal chaotic attractor into a wing.

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