Walter J. Freeman
Journal Article e–Reprint
Nonlinear Neural Dynamics in Olfaction as a Model for Cognition
W. J. FREEMAN
Springer Series in Brain
Dynamics 1
Edited by Erol Basar
Springer–Verlag
Berlin Heidelberg 1988
1 Introduction
The forebrain of primitive
vertebrates is so heavily devoted to olfaction that for half a century
investigators were misled into considering the function of the hippocampus as
being exclusively olfactory. For example, the anterior third of the forebrain
of the tiger salamander forms the bulb, the medial third is hippocampus, and
the lateral third comprises the piriform. and striato–amygdaloid complex
(Herrick 1948). According to Herrick, a transitional zone in the mantel
receives thalamic axons that convey input to the forebrain from all other
sensory systems. He proposed that with the expansion and increasing dominance
of these other systems, the brain expanded by adding new parts while preserving
the topology of connections of those parts already existing. This view has
survived to the present with modifications; it is as if, seeing that olfaction
was a success, other systems moved in and co–opted the machinery of the
forebrain. Olfaction remains the simplest among the sensory systems. For this
reason, if for no other, the study of sensation and cognition might well begin
with the sense of smell. But there are three other good reasons: the parallels
that exist between olfaction and other senses in their psychophysics, in the
dynamics of the masses of neurons comprising them, and in the types of neural
activity that they generate.
2 Psychophysics
The olfactory system resembles
other sensory systems in consisting of a surface array of receptors of multiple
kinds that project in parallel to arrays of central neurons. Some examples of
stimuli that are comparable to an odorant are the sight of a constellation such
as Orion in the winter sky, the feeling of putting on a coat that is not the
correct size, and the sound of a tone that allows immediate identification of
the instrument being played – a piano, oboe, etc. These operations are
rapid, spatial, and global, and they depend on past experience. The information
is expressed by spatial relationships among activated and equally importantly
nonactivated receptors, without reference to simple geometric forms. The time
frames are longer than that of an action potential but shorter than a heart
beat; according to Efron (1970), on the order of 0.1 s.
All of these
systems are legendary for their sensitivity and at the same time for their
stability and broad dynamic range, qualities that engineers often find to be
antithetical. Olfactory sensitivity lies in part in the regenerative molecular
feedback mechanisms of single cells (Lancet et al. 1985) such that a single
odorant molecule may trigger a train of action potentials in a receptor cell.
However, sensitivity is also provided, especially in macrosmatic animals, by
the immense numbers of receptors. In the cat, for example, there are in the
order of 108 receptors on each side, a numerosity enhances the
likelihood of capture of molecules in turbulent air passed over the turbinate
bones. Herein lies a major difficulty in understanding olfaction, which Lashley
(1950) identified in vision as the problem of stimulus equivalence. Supposing
that there might be in the order of 10–100 types of receptor, then there
must be 1–10 million of each type. If an odorant can be identified
repeatedly at concentrations ranging over 3–5 orders of magnitude, and if
the lowest concentration involves stimulation of 10–100 receptors, how is
an invariant constructed by the brain for an odorant over multiple trials, when
the spatial pattern of excited receptors is never twice identical? The same
type of problem obviously occurs in visual recognition of faces or signatures
and auditory recognition of voices or words.
Sensitivity in olfaction is enhanced by
experience under reinforcement and is disenhanced without it. Most of us have a
limited repertoire of about 16 odorants under absolute discrimination, but the
number can be increased without limit by sustained practice (Cain and Engen
1977). We can recognize some odors that were once important to us at intervals
over many years in a flash flood of vivid associative memories that impel us to
action. These are basic properties that olfaction shares with all other senses,
far transcending in importance the decomposition of stimuli into lines, planes,
and spectral peaks.
3 Neural Dynamics:
Nonlinearity
I propose here that all of
these properties inhere in the bulb in a single, comprehensive. nonlinear
operation. The main bulbar constituents are large numbers of densely
interconnected excitatory neurons (the mitral and tufted cells) and inhibitory
neurons (the granule cells). The receptor input (Fig. 1) is spatially coarse–grained
into segments corresponding to glomeruli which form the bulbar equivalent of
cortical columns with a mean segment width of about 0.25 mm. There are about
2000 glomeruli in each bulb of the rabbit. The several types of periglomerular
interneurons in the outer layers of the bulb perform various janitorial tasks
of input dynamic range compression, automatic volume control, spatial contrast
enhancement, clipping, holding, and dc offset or bias regulation, among others
(Freeman 1975). The negative feedback relation between the mitral and granule
cells (Rall and Shepherd 1968) establishes a neural oscillator that receives
its input through each glomerulus. These oscillators are coupled by mutually
excitatory axosomatic synapses broadly over the bulb (Nicoll 1971) and by
mutually inhibitory interactions through cellular mechanisms not yet clearly
identified. Their output under coupling is at a frequency in the gamma range of
35–90 Hz, determined in the main by the passive membrane time constants
(about 5 ms) and by the gains in the three types of feedback loop. Because of
the widespread coupling, the EEGs from all parts of the main bulb at all times
have a common waveform and everywhere a common instantaneous frequency (Freeman
1986).
These
oscillators are inherently nonlinear. The nonlinearity stems from the voltage–dependent
nonlinearity modeled for the action potential of nerve membrane by the Hodgkin–Huxley
equations (Freeman 1979a). In the neural ensemble, it emerges as a sigmoidal
function (Fig. 2) that relates pulse density (pulses per second per unit volume
of the ensemble) to the density of excitatory dendritic current at the trigger
zones. The curve is asymptotic to zero pulse density with inhibitory
postsynaptic potential (IPSP) current and to a maximum for the ensemble with
excitatory postsynaptic potential (EPSP) current. Two processes combine to give
this shape. One is the exponential increase in tendency to fire with increasing
depolarization (the sodium permeability or m–factor in the Hodgkin–Huxley
equations). The second is the collection of metabolic, restorative,
accommodative, and hyperpolarizing processes that establish the upper limit on
firing rate, both on the long–term firing of single neurons and, by the
ergodic hypothesis, on the entire ensemble over the short term. The
nonlinearity is static, as distinct from the time–varying linear
relationship that holds between membrane potential and firing rate for
regularly firing single neurons. This is because neurons spend 99% of their
lifespan below threshold, and because the firing pattern of each neuron closely
resembles a Poisson process unrelated to those of its neighbors.
The nonlinear
function is determined experimentally by calculating the pulse probability of
mitral cell firing conditional on the EEG amplitude. The calculation is repeated
for each EEG sample at 1 ms digitized intervals forward and backward in time
± 25 ms, in order to allow for the time lags in the neural oscillator.
The procedure also serves to demonstrate that the firing probability of each
mitral cell oscillates at the common EEG frequency, and that the modulation
amplitude in firing rate co–varies with the peak–to–peak
amplitude of EEG oscillation. Mitral cell firing is statistically closedly
related to the EEG at all times and at each point of the bulb.
Fig. 1. A flowchart of
activity in the olfactory system. Each layer is organized into a sheet of
neurons. The state variables are defined for the axonal and dendritic modes in
the two surface dimensions. They are discretized at intervals corresponding to
the spatial coarse–graining by the glomeruli. Interactions occur
laterally in each laver. The primary olfactory nerve provides for topographic
projection of the input, whereas the lateral olfactory tract provides for
spatial integration of the output. (From Freeman 1983)
The
nonlinear function for each bulbar ensemble is under centrifugal control. The
shape of the sigmoid curve is retained, but the steepness is subject to
increase, along with an increase both in mean and maximal firing rates. The derivative
of the function represents the nonlinear gain of each local ensemble. The
maximal gain is always displaced to the excitatory side. In animals under
increased arousal or motivation, the gain is increased and the displacement to
the excitatory side is extended, along with the increase in mean firing rate.
The centrifugal input is most likely the cholinergic projection to the outer
layers of the bulb. On the peripheral side, any receptor input excites the bulb
and thereby raises its mean firing rate and its instantaneous gain. The curve
is fixed but the operating point changes. Owing to the surge of receptor input
with each inhalation, the bulb tends to undergo a recurrent increase and
decrease in gain with the respiratory cycle.
Because of the
bilateral saturation, the sigmoid curve is the most important mechanism
providing for the stability of the bulbar mechanism (Freeman 1979b). The same
curve also provides for its remarkable sensitivity, in the main because of the
mutually excitatory feedback loop. Excitation of one subset excites another
which re–excites the first, now in a more sensitive state, so that a
regenerative increase in activity can occur. However, the negative feedback
gain is also increased, so that instead of runaway excitation, a burst of
oscillation appears. It begins during inhalation and ends during exhalation,
and it is seen only in aroused, motivated animals (except occasionally in light
stages of anesthesia, and then in an abnormal frequency range).
Fig. 2. Top, three examples of a curve fitted to
statistical data showing conversion of dendritic current density to axonal
pulse density. Bottom, derivatives of the three curves that give the nonlinear
gain. Triangles, resting or equilibrium values. With increasing current amplitude
there is a coupled increase in pulse density and in gain. (From Freeman 1979a)
4 Neural Dynamics: Spatial
Properties
Studies of the spatial
patterns of these bursts manifested in the EEG have been made in rabbits with
arrays of 64 electrodes chronically implanted over the lateral surface of the
bulb. The EEG shows no dependence on novel odorants presented to naive animals,
other than nonspecific chances associated with orienting responses. The spatial
patterns of amplitude and phase modulation of the burst frequency vary within
narrow limits about stereotypic mean patterns that are as characteristic for
each individual as a handwritten signature. Under classical conditioning to
respond differentially to two ordors (Viana di Prisco and Freeman 1985) one
reinforced [conditioned stimulus (CS) + ] and the other not (CS – ) two
new spatial patterns of amplitude emerge (Fig. 3), one for each CS. They are
present only when the one or the other CS is present (Freeman 1986). For this
demonstration, the EEG must be filtered with digital filters designed to
conform to the spatial and temporal passbands of the granule cell contribution
to the EEG (Freeman 1986). The resultant patterns serve together with
discriminant analysis to classify correctly, on average, 82% of EEG bursts
sampled during control and test odor periods (Freeman 1986: Freeman and Viana
di Prisco 1986). These patterns cover the entire array and, by inference from
surface EEG phase gradients (Freeman 1986) and depth recording (Bressler 1984),
the entire main bulb. The information density over the bulb is spatially
uniform to within ± 5% (SD) of its mean, as measured by its value for
correct classification of bursts.
Fig. 3. Density plots (seven
levels in descending order of amplitude # * + = – . ) of EEG activity. Upper
frames, means
and SDs of amplitudes (Chaos refers to the disorderly bursts not subject to
classification in respect to odors). Lower frame, amplitudes normalized by
channel and by group, with those correctly classified on the left and those incorrectly
classified by discriminant analysis on the right. Bottom row, patterns reconstructed from
factor scores and loadings that were used for classification. (From Freeman
1986)
The results show
that insofar as the EEG is concerned, the bulb has the capability of responding
selectively to odorants, but only in aroused animals that are trained to detect
and respond to the test odors. This is in striking contrast to the results from
unit studies in anesthetized or immobilized animals, which show selective
responding of single neurons to some odors and not others, irrespective of
training (e.g., Moulton 1976). Studies of metabolic activity with 2–deoxyglucose
show that different patterns of radiographic density in the glomerular layer
result from presentation of different odors (e.g.. Lancet et al. 1982). These
studies still lack proper controls for individual variation. The method allows
only one odor for each animal; the EEG method shows foci of high amplitude
activity that are similar to the high–density metabolic foci in size,
shape, and location. but the degree of variation in EEG pattern between
individuals exceeds that between odorants for each individual. Still, it is
reasonable to conclude that input to the bulb from receptors establishes local
regions of activity specific to an odor, and the output of the bulb is a global
pattern involving all bulbar neurons, provided that the animal has been
trained. Otherwise the global bulbar response is not spatially or temporally
coherent or reproducible.
This
transformation of local input to global output that incorporates past
experience is the key to bulbar function. It is best understood by description
in terms of nonlinear dynamics (Garfinkel 1983). A set of distributed, coupled,
nonlinear oscillators has an infinite number of ways of performing, but within
certain conditions of input and interaction strengths it tends to enter a
definable state of activity and stay there until perturbed or modified. If
under repeated perturbation it tends always to return to the same state, the
system dynamics is said to have, or be governed by, an attractor. Attractors
fall into three classes. The simplest is that of equilibrium; this occurs in
the bulb only under deep anesthesia or in death. Periodic oscillation
characterizes the limit cycle attractor: this appears in the EEG during bursts
with inhalation. The most complex type is called the strange or chaotic
attractor; its manifestation is nonperiodic activity that may appear to be
random, of the sort that characterizes the resting EEG in nonmotivated animals
and also the low–level EEG activity during exhalation.
Switching from
one attractor to another is called a "state change" or
"bifurcation"'. Its occurrence requires a parametric change in the
system. Bulbar input provides for this by virtue of the nonlinear gain increase
with receptor input during inhalation. The state change is from low–amplitude
chaos to a high–amplitude spatially coherent limit cycle, and then back
again. Order emerges from chaos and collapses with each cycle of respiration.
There may be indefinitely many attractors of each type. Each is characterized
by a set of parameter values and by a basin defined by a domain of input. The
evidence suggests that a limit cycle attractor may form for each odorant that
an animal is trained to respond to.
I believe that a
limit cycle attractor is formed in the following way. On each inhalation of an
odorant, the subset of the receptors that is sensitive to the odorant
coactivates a subset of mitral cells. These are interconnected by excitatory
axosomatic synapses that are bidirectional (Willey 1973). In accordance with
Hebb's rule (Hebb 1949; Viana di Prisco 1984), these synapses are strengthened
under coexcitation, provided that a reinforcing stimulus is paired with the
odorant. Reinforcement activates neurons in the locus coeruleus, thus releasing
into the bulb (and elsewhere) norepinephrine that enables the synaptic change
(Gray et al. 1984). With repeated inhalations in the same and sequential
trials, the odorant is delivered by turbulent flow in the nose to an ever–changing
fraction of the subset of sensitive receptors, which leads progressively to the
ultimate inclusion of all those mitral cells to which they project into a nerve
cell assembly. These strengthened, mutually excitatory connections give the
property to the assembly that, if any fraction of the sensitive receptors
receives the odorant, their input to the bulb excites the entire assembly in a
stereotypic manner (Freeman 1979c).
At once this
constitutes figure completion, generalization over equivalent stimuli, and
sensitization specific to a repeatedly reinforced class of stimulus. Computer
simulations (Freeman 1979b) have shown that an increase of 40% on average in
synaptic strength may increase the sensitivity of the bulb to a particular
odorant by as much as 40,000 times above the basal or naive level, because of
the combination of mutual excitation and the nonlinear gain. After the
completion of training, the subset of receptors activated during the training
defines the basin of the attractor, and the nerve cell assembly of mitral cells
determines the spatial structure of the limit cycle oscillation, which extends
well beyond the assembly to involve the entire bulb. In principle, we can show
how one odorant molecule can shape the activity of several hundred thousand
second–order neurons.
I conceive the
bulb as carrying a repertoire of learned limit cycle attractors, one for each
odorant previously reinforced. Each is distinguished by its input basin with
respect to receptors and by the spatial amplitude modulation pattern of its
output. Random access is facilitated by the chaotic basal state, which keeps
the bulb far from equilibrium and ready to move rapidly to any region of
optimal convergence. The steadfast spatial pattern of bursts in the control
state, in which no reinforced odor is given indicates that an attractor exists
for the background odor complex as well, and that bulbar output then signals
the status quo. If a novel odor is given. the result is suppression of orderly
burst activity and the appearance of broad–spectrum, spatially irregular,
and nonreproducible bursts. Commonly, the highest peak of their multiply peaked
spectra is at a frequency about half that of the sharply tuned frequency of the
orderly bursts. I call these bursts "disorderly" or
"chaotic". The prepyriform cortex to which the bulb projects responds
to input as a tuned oscillator with spectral resonances around 18–24 Hz
and 40–70 Hz (Freeman 1975). This suggests that the lower transmission
frequency of the chaotic bursts can signal the failure of the bulbar mechanism
to converge to a limit cycle attractor, and that repeated failures can lead to
either of two outcomes: habituation if there is no reinforcement which updates
sensitivity to a new status quo, or formation of a new limit cycle attractor
under reinforcement. In other words, the bulbar mechanism provides a novelty
detector without requiring an exhaustive search through information stored in
the bulb.
Although the
bulb has numerous specialized features not found elsewhere in the brain, these
are not responsible for its main properties. At base it consists of a sheet of
interconnected excitatory and inhibitory neurons with parallel input and
output. This is an elementary description of neocortex as well. The static
nonlinearity is a generalizable property of axonal membrane to be expected for
every large ensemble in the cerebral cortex. The time and space constants are
common to many, if not most cerebral neurons. Hence the same basic dynamics can
be expected to exist in all parts of the cerebral cortex.
I infer that
odorant information is conveyed to the bulb by action potentials on particular
receptor axons and that excitation is established and integrated among local
subsets of mitral cells having apical dendrites within a limited number of
glomeruli that correspond to neocortical columns. Following bifurcation, the
entire bulb, comprising roughly 1 cm2 of cortical tissue, goes to a
limit cycle attractor in the basin selected by the input. The output is global;
the information is conveyed by action potentials on mitral axons, but it is in
the form of a macroscopic pulse density function that is continuous in time and
the two surface dimensions. The information is imposed as spatial amplitude
modulation (in the surface dimensions as distinct from the time envelope) of
the limit cycle carrier oscillation that is common to the entire bulb. Each
event lasts in the order of 75–100 ms and repeats at the respiratory rate
of 1–7 Hz. At the macroscopic level, each event can again be discretized
into the surface grain of the glomeruli and the time frame of the burst; that
is, olfaction can be treated as a sampled data system analogous to a digital
graphic display.
The intrinsic
state variables of a model for this system must correspond to the active states
of pools of like neurons, which Sherrington identified as their central
excitatory states (CES). For this reason, the proposed view might be described
as neo–Sherringtonian. These activities are conveyed in local concentrations
of action potentials, transmitter substances, and dendritic currents. They are
manifested to observers in the forms of unit activity and electromagnetic field
potentials. In all instances, the measurements of these observables must be
properly filtered, averaged, and otherwise transformed in order to bring them
into conformance with the CES, and they must be assigned to the proper elements
in the model; for example, in the bulb, the EEG should be assigned to the
granule cells and unit activity at the appropriate depth should be assigned to
mitral cells.
The parallels to
other sensory systems are straightforward. Information is conveyed by action
potentials on thalamocortical axons and is established in local regions
corresponding to columns, with different kinds of information being established
at the microscopic level in each of the multiple cortical subareas comprising a
sensory projection area. The neurons onto which the afferent activity is
projected consist of excitatory and inhibitory neurons that are known to be
densely interconnected by negative feedback and mutual excitation and can be
inferred to have mutual inhibition as well. The crucial step for integration in
perception may be the bifurcation of the interactive neural mass from a low–level
chaotic attractor to a learned limit cycle attractor, such that the output of
an extensive area of cortex at the macroscopic level might convey information
on the whole in the spatial modulation of the amplitude of the limit cycle
frequency. Again. input is local, output is global, and in analogy to the
hologram. all parts of the output reflect all parts of the input.
Investigation of
this hypothesis is likewise straightforward. The requisite carrier and gating
frequencies respectively in the high beta and gamma ranges and in the theta and
alpha ranges have been observed in most areas of neocortex. In visual cortex, during
alpha suppression, the sequence of bifurcations requisite for trains of bursts
might be provided by saccades. The steps that are needed to test the hypothesis
are (1) the detailed spectral characterization of these activities, including
use of complex demodulation over extended time series of the EEG; (2) the
identification of the sources and sinks of the electric currents underlying
these spectral peaks; (3) the assignment of these activities as states of
variables of identified types of neurons in the cortex (4) measurement of the
open loop time and space constants under deep anesthesia (Freeman 1975); (5)
establishment of the spectral and spatial domains of neocortex over which
commonality of wave form holds, such that chaotic or limit cycle attractors can
be sought: and (6) behavioral analysis to determine the dimensions of the
activity that relate to the stimulus and response variables selected for
testing. Some progress has already been made in relating information content of
visual and auditory stimuli to the waveforms of event–related potentials
from neocortex. According to the present hypothesis, these correlations are
adventitious and secondary, because the information relating to content is to
be sought in the spatial dimensions, while the time courses of events are
expected to reflect primarily the neural operations being performed on that
information (Freeman and Schneider 1982).
None of these
six steps is trivial; each may require several years to be brought to fruition.
The outcome will be exceedingly important, because these kinds of information
are essential to devise, evaluate, and improve macroscopic models of the
distributed nonlinear dynamics of the forebrain.
In conclusion,
the essence of cognition lies in forming and testing expectations based on past
experience. In science it takes the form: if I do X, I expect A or B or the
unexpected. Each outcome has predictable consequences. In rabbit olfaction it
takes the form: if inhalation, then either status quo (the background), odor A,
odor B, or an unexpected odor. Each inhalation is the action of a pattern
generator or limit cycle attractor in the brain stem respiratory nuclei; each
neural response is mediated by limit cycle attractors in the bulbs. I postulate
that licking and sniffing are likewise mediated by limit cycle attractors in
motor systems, whose basins receive the output of the bulbs. Basically this is
a simple model of simple conditioned reflexes, but it tells us what to look for
and how to look for it, as we try to understand how the brain synthesizes a
percept from diverse sensory detail in the literal twinkling of an eye or the
wriggle of a nose.
Summary
Neurons in cerebral cortex interact synaptically by
mutual excitation, mutual inhibition. and negative feedback. Typically the
negative feedback connections are locally dense, leading to the formation of
local oscillators corresponding to columns. They are interconnected by mutually
excitatory connections over large cortical areas. An appropriate model of
cortex is a sheet of distributed coupled oscillators: observation is performed
with arrays of surface EEG electrodes.
The dynamics of
such systems are shaped by tendencies under perturbation to converge to stable
states that are identified with attractors of three kinds. An equilibrium attractor is manifested in
cortex by a steady state under deep anesthesia–, a limit–cycle attractor is manifested by
regular oscillation, and a strange attractor is manifested by chaos that appears to be
random activity. Transition (bifurcation) from one attractor to another is
imposed by a parametric change of the model or cortex.
The EEG of the
olfactory bulb at rest appears chaotic. Inhalation excites the bulb, causing a
parametric increase in negative feedback gain; the bulb bifurcates to a limit–cycle
state. In the control condition of breathing air, the EEG spatial pattern is
stereotypic for the background odor complex. With training to discriminate
odors, a new spatial pattern appears with each odor, manifesting a learned
limit–cycle attractor. These patterns appear to cover the entire bulb;
input is local and output is global. The integration of a stimulus with past experience
takes less than 0. 1s. Other sensory systems have similar properties; therefore
bulbar dynamics may provide a useful model to explore preattentive processing
in vision and other cognitive operations in the neocortex.
Acknowledgement. This work was supported by a
grant MH06686 from the National Institute of Mental Health.
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