Learning and pattern recognition by populations
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 learning 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 Figure 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 (Figure 3). The increase in synaptic strength with learning 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 learns to discriminate it. The assembly consists of a small subset of cells that were co-excited on one or more learning trials on which reinforcement was given. This combination of conditions strengthens the synaptic connections between them (Gray, Freeman and Skinner, 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 learning 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 (Figure 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 the 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 (Figure 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 (Figure 10, lower frame). The EEG may often seem to be close to periodic, but its spectrum is broadly distributed about certain preferred peaks, and it 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 (Figure 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 Figure ll 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, this one 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 de-stabilized. 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.