mesopic neurodynamics - from neuron to brain

7/31/2019 Mesopic Neurodynamics - From Neuron to Brain

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J. Physiol. (Paris) 94 (2000) 303322

2000 Elsevier Science Ltd. Published by Editions scientifiques et medicales Elsevier SAS. All rights reserved

PII: S0928-4257(00)01090-1/REV

Mesoscopic neurodynamics: From neuron to brain

Walter J. Freeman*

Department of Molecular and Cell Biology, Uni6ersity of California, Berkeley, CA 94720-3200, USA

Received 21 June 2000; accepted 7 August 2000

Abstract Intelligent behavior is characterized by flexible and creative pursuit of endogenously defined goals. Intentionality is a keyconcept by which to link neuron and brain to goal-directed behavior through brain dynamics. An archetypal form of intentionalbehavior is an act of observation in space-time, by which information is sought for the guidance of future action to exploreunpredictable and ever-changing environments. These acts are based in the brain dynamics that creates spatiotemporal patterns ofneural activity, serving as images of goals, of command sequences by which to act to reach goals, and of expected changes in sensoryinput resulting from intended actions. Prediction of the sensory consequences of intended action and evaluation of performance is byreafference. An intentional act is completed upon modification of the system by itself through learning. These principles are wellknown among psychologists and philosophers. What is new is the development of nonlinear mesoscopic brain dynamics, by which thetheory of chaos can be used to understand and simulate the constructions of meaningful patterns of neural activity that implement

the process of observation. The design of neurobiological experiments, analysis of the resulting data, and synthesis of explanatorymodels require an understanding of the hierarchical nature of brain organization, here conceived as single neurons and neuralnetworks at the microscopic level; clinically defined cortical and subcortical systems studied by brain imaging (for example, fMRI) atthe macroscopic level, and self-organizing neural populations at an intermediate mesoscopic level, at which synaptic interactions createnovel activity patterns through nonlinear state transitions. The constructive neurodynamics of sensory cortices, when they are engagedin pattern recognition, is revealed by learning-dependent spatial patterns of amplitude modulation and by newly discovered radiallysymmetric spatial gradients of the phase of aperiodic carrier waves in multichannel subdural EEG recordings. 2000 Elsevier ScienceLtd. Published by Editions scientifiques et medicales Elsevier SAS

EEG / intentionality / mesoscopic brain dynamics / perception / reafference

1. Introduction: intentionality provides the contextfor brain dynamics

The salient characteristic of animal and humanbehavior is its directedness toward goals that be-come apparent as sequences of actions bring theobserved person or animal toward a situation orcondition that can be identified as capture ofsomething of value, such as food, drink, a mate, orshelter from danger or discomfort. In a broad viewthese goals are set by and consistent with basicbiological requirements for survival and reproduc-tion, but what is more interesting is the manner inwhich the specific behaviors displayed by organ-isms that satisfy these requirements emerge fromwithin. The acts are not initiated by events in theirenvironments; they are designed, implemented,

and adapted to rapidly changing conditions in theenvironments by emergent dynamics in the organ-isms, in the vertebrates primarily through the self-organizing neurodynamics of their forebrains.

These goal-directed behaviors have been known tophilosophers as intentional since the work ofThomas Aquinas [1], who introduced the term todescribe the process by which humans and animalslearn about the world by adapting and assimilatingto the specific conditions of their environments. Aprincipal problem was how the stimuli comingfrom the environment could lead to the acquisitionof knowledge. He concluded that single events inthe material world are not knowable, and thatknowledge comes only through abstraction andgeneralization from the phantasmata of raw sen-sory impacts.

This conclusion is immediately relevant to ex-perimental findings that are reviewed here. In par-ticular, when the neural activity evoked by astimulus is observed in the brains of small animalsthat have been trained to respond behaviorally, itis clear that the microscopic patterns forced bysingle stimuli are not retained, and that they leadto replacement by a mesoscopic activity pattern[14] constituting a generalization over the class ofstimuli [12]. This review is devoted to a descriptionof this process of generalization and related opera-tions of brain dynamics in the context ofintentionality.

* Correspondence and reprints.

E-mail address: [email protected]

(W.J. Freeman).


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W.J. Freeman / Journal of Physiology 94 (2000) 303322304

1.1. A brief o6er6iew of intentionalitygo6erning an act of obser6ation

The first step in pursuit of an understanding of

the neurodynamics of intentionality is to ask, whathappens in brains during an act of observation?This is not a passive receipt of information fromthe world. It is a purposive action by which anobserver directs the sense organs toward a selectedaspect of the world and selects from the resultingbarrage of sensory stimuli the small fraction that isrelevant to the intent of the action. The concept ofintentionality was used by Aquinas to describe thisaction-perception cycle in both animals and hu-mans, while distinguishing it from volition, whichinvolved awareness of the ethical dimensions ofaction. Three salient characteristics of intentional-ity that can be inferred from his work are (a)

intent, (b) wholeness, and (c) unity [12].(a) Intent (from the Latin word intendere,

meaning thrusting forth) comprises the endoge-nous initiation, construction, and direction of be-haviors into the world, followed by learning fromthe consequences of the behaviors. It is the processthat governs Merleau-Pontys [39] intentional arc.It emerges within brains. Agents including hu-mans, animals and perhaps autonomous robotsselect their own goals, plan their own tactics, andchoose when to begin, modify, and stop sequencesof action. They learn about their environments byassimilation (from the Latin word adequatio,meaning approaching equivalence, like the way a

hand is shaped to the form of a cup for drinking).Humans at least are subjectively aware of them-selves acting, but consciousness is not a necessaryproperty of intention; in any case, studies of scalpevent-related potentials by Walter [58], Kornhuberand Deecke [36], and Libet [37] indicate that hu-mans become aware of their intentions only in thecourse of unfolding their intended behaviors afterthey have been initiated, so that intentional actionprecedes consciousness.

(b) Unity is inherent in the combining of inputfrom all sensory modalities into gestalts, and in thecoordination of all parts of the body, both muscu-loskeletal and autonomic, into adaptive, flexible,yet integrated movements through the motor neu-rons in what Sherrington in 1906 [41] called thefinal common path. Subjectively, unity appears inthe awareness of self and its feelings of emotion,but again this is not prerequisite for unity, in thatintentional actions are commonly not conscious.

(c) Wholeness is revealed by the orderly changesin the self and its behavior that constitute the

development, maturation and adaptation of theself, within the constraints of its genes or designprinciples, and its material, social and industrialenvironments. Subjectively, wholeness is revealed

in the remembrance of self through a lifetime ofchange. The influences of accumulated and inte-grated experience on current behavior are not de-pendent on explicit recollection and recognition. Inbrief, the study of intentionality can be directedtoward understanding the neural mechanisms bywhich goal states and goal-directed behaviors areconstructed, executed, and evaluated, while avoid-ing invoking the subjective processes of conscious-ness, awareness and emotion. The prototype ofintentional behavior can be found in a directed,purposive act of looking, listening, touching, orsniffing, irrespective of whether the observer, ani-mal or human, is or is not conscious of the act and

its results.These aspects of intentionality correspondroughly to three current uses of the term. Inpsychology it means purpose, as in He intended todo an experiment. Intent is often conflated bypsychologists with motivation, but not so bylawyers, who distinguish between intent as thatwhich is to be done, and motive as the reason forthe intent. In medicine and surgery, it means themodes of healing and re-integration of the bodyafter damage; healing by first intention leaves aclean scar; healing by second intention is accompa-nied by the flow of pus. These uses began early inthe 14th century, when the philosophy of Aquinas

was gaining wide acceptance, and have persisted.Among medieval philosophers knowledge by firstintention was immediate sense experience, andknowledge by reflection was by second intention.In the 20th century analytic philosophers use in-tentionality to mean the way in which beliefs,thoughts and images (mental representations) areconnected with (about) objects and events in theworld. This is also known as the symbol-ground-ing problem: how are representations in brains andcomputers to be attached to the things they repre-sent? The Aquinian view avoids and disavowsrepresentations as Platonic forms, and insteadposits forms that are created intentionally by the

imagination. These forms and the mesoscopicdynamics by which they are created are the objectsof our search in animal brains.

1.2. Sensorimotor and limbic cortices areorgans of intentional beha6ior

Brain scientists have known for over a centurythat the necessary and sufficient part of the verte-


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W.J. Freeman / Journal of Physiology 94 (2000) 303 322 305

brate brain to sustain minimal intentional behav-ior is the medioventral forebrain, including thosecomponents that comprise the external shell of thephylogenetically oldest part of the forebrain, the

archicortex and paleocortex, the deeper lying nu-clei with which the cortices are connected, and theolfactory bulb that receives direct sensory inputfrom olfactory receptors. These components sufficeto support remarkably adept patterns of inten-tional behavior in dogs after all the newer parts ofthe forebrain were surgically removed by Goltz[23], and in rats with neocortex chemically inacti-vated by spreading depression as shown by Bureset al. [3]. In contrast, intentional behavior isseverely altered or absent after major damage tothe medial temporal lobe of the basal forebrainbilaterally, as manifested for example inAlzheimers disease.

Phylogenetic evidence comes from observing in-tentional behavior in salamanders, which have thesimplest of the existing vertebrate forebrains [29,40]. The three main parts are sensory (which, as insmall mammals, is predominantly olfactory), mo-tor, and associational (figure 1).

These parts can be judged to comprise the lim-bic system in all vertebrates, but salamander havevirtually none of the add-ons found in brains ofhigher vertebrates, hence the simplicity. The asso-ciational part contains the primordial hippocam-pus, which is identified in higher vertebrates as thelocus of the functions of spatial orientation (thecognitive map of Tolman [51] and OKeefe andNadel [40]) and temporal integration in learning

Figure 2. This diagram of brain state space maps the multiplefeedback loops that support the intentional arc. In mammalsthe entorhinal cortex is a key structure. Its most immediateinteraction is with the hippocampus, involving temporal inte-gration. Interactions are sustained through longer loops with

all of the primary receiving areas, supporting reafference [57],and with motor systems in the brainstem, cerebellum andneocortex, subserving motor control. Longer loops passthrough the body in proprioception, and through the environ-ment in intentional action.

(the organization of long term and short termmemory), operating through the septal nuclei andhypothalamus to control the autonomic and en-docrine systems, and through the amygdaloid andstriatal nuclei to control the musculoskeletal ap-paratus. These processes are essential for inten-tionality, because intentional action takes placeinto the world, and even to execute the simplestaction, such as searching for food or evading apredator, an animal must make decisions and planactions based on where it is with respect to itsworld, where its prey or refuge is or is predicted tobe, and what its spatial and temporal progress isduring sequences of attack or escape. It mustmobilize its cardiovascular, respiratory, andmetabolic systems to meet the demands on itsmuscles, and it must have feedback to assay itsprogress and resources at each step of the action.Multiple feedback loops interconnect numerousstructures with each other inside the brain, outthrough the body, and out through the environ-ment (figure 2). These connections support theflow of neural and behavioral activity that is re-quired for the neurodynamics of intentionality.

2. Observation of mesoscopic statesby recording multichannel EEGs

The crucial question for neuroscientists is, howare the patterns of neural activity that sustain

Figure 1. This schematic illustrates the sensory, motor, and

associational components of the right hemisphere (seen fromabove) of the simplest extant vertebrate brain, found in thesalamander. The bidirectional connections between these fore-brain subdivisions provide for the macroscopic interactions thatsupport the neurodynamics of intentionality: goal formation,action, perception, and learning from the sensory consequencesof the actions. These components are the prototype of thelimbic system, which in mammals are typically buried byexuberant growth of other add-on structures that operate inand in concert with the limbic system.


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intentional behavior constructed in brains prior toperception? An answer is provided by studies ofelectrical activity, the electroencephalogram (EEG)recorded from the pial surfaces of primary sensory

areas in animals trained to respond to conditionedstimuli [2, 9, 13, 14, 20, 53].

2.1. Mesoscopic neurodynamicsof the olfactory system

The most detailed studies have been devoted tothe olfactory system [14]. The main findings are asfollows. Arrays of closely spaced electrodes (88with 0.5 mm spacing giving a window of 44mm) placed on the olfactory bulb, nucleus, orprepyriform cortex show continual backgroundEEG activity having two main components. In thegamma range (2080 Hz in the rabbit) there are

bursts of oscillations at continually varying fre-quencies, which gives spectra of the EEG the formof 1/fh (where h=291 and log power decreaseslinearly with approximately the square of log fre-quency). The bursts recur at the rate of respira-tion, generally in the theta range (27 Hz),accompanied by a slow wave in the EEG that hasa negative peak following inhalation and usuallycorresponding to the onset of a burst (figure 3).The same wave form is found on all electrodeswithin the same structure of recording, but withdiffering spatial patterns of phase and amplitude.

Unlike the time series of the EEG, which showsno relation to odorants, the spatial pattern of

amplitude modulation (AM) changes significantly

under both aversive and appetitive classical condi-tioning to odorants [21]. Changes in AM patternare progressive over both serial [19] and concomi-tant discriminative conditioning [55], so that when

a new AM pattern forms with learning, or whenthe significance of an odorant pair is changed byreversing the contingency of reinforcement, all pre-existing AM patterns change slightly but signifi-cantly [19, 20]. The AM patterns lack invariancewith respect to conditioned stimuli, as shown mostclearly under serial conditioning; when an odorantis re-introduced after conditioning to interveningstimuli, the AM pattern changes to a new formand does not recur to a pre-existing form (figure4).

An interpretation of these findings is that witheach inhalation the olfactory receptors excite thebulb and depolarize the bulbar neurons, giving rise

to the slow respiratory wave, in which surfacenegativity manifests excitation. The bulbar neu-rons have a nonlinear input-output relation knownas the sigmoid curve (figure 5), for which theslope shows their strength of transmission orgain. Excitation increases the gain to a thresholdwhere the synaptic interaction in the bulb increasesexplosively. This increase is analogous to thechange in state of a single axon, which is forced bydendritic current across its threshold. That is, theburst of cortical activity is comparable to theaction potential of a neuron. In each case thesystem undergoes a state transition from a recep-tive (diastolic) mode to a transmitting (systolic)

mode. Most of the synapses received by bulbar

Figure 3. EEGs are shown from electrodes in the olfactory bulb and prepyriform cortex, and respiration is shown by a pneumographtracing, during a trial in which an odorant conditioned stimulus is followed by a weak electric shock to the cheek in classical aversiveconditioning. The surge of receptor input on inhalation excites the bulb (negative slow wave) and destabilizes it, leading to the gammabursts.


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Figure 4. A representative set of 64 traces is shown from an 88 array (44 mm) placed on the surface of the olfactory bulb (leftis anterior), filtered at 2080 Hz to display the gamma range. The contour plots of amplitude contrast the differences in AM patternsbetween the control and test periods prior to and during odorant delivery. The changes between trials set 2 weeks apart demonstratethe context dependence and lack of invariance of AM patterns with respect to the control and conditioned stimuli, which wereunchanged.

neurons come from other bulbar neurons and notfrom receptors, which indicates that in the regener-atively excited state the dominant input to eachbulbar neuron is by synapses from other bulbarneurons instead of receptor axons. These synapsesfrom internally originating axons are the ones thatare modified by learning [7, 8].

These findings lead to the conclusion that theAM patterns are determined not by immediatereceptor input but by cumulative synaptic changeswith experience. The construction is not by recallof stored patterns but by pattern formation in adistributed nonlinear system with connections thathave been modified through learning. The mannerin which this take place involves hierarchical or-dering of neural activity between microscopic andmesoscopic levels having differing time and spacescales. Central neurons are selectively activated bysensory receptors and driven to generate micro-scopic activity in the form of trains of actionpotentials (pulses) on their axons. These and allother neurons in the bulb by their synaptic interac-tions form a population that binds their activityinto a mesoscopic pattern, which expresses theexperience with the stimulus and not the presenta-tion of the stimulus.

The significance of this process of AM patterncreation can be seen in terms of the variability ofthe selection of olfactory receptors that are excited

on each inhalation of the same odorant. Owing tothe small sample with each sniff and to the turbu-lence of air flow in the nose, the receptor patternmust differ with each trial, and it cannot beknown, either by the animal or by the experi-menter. The action of the bulb is to create bysynaptic changes with learning a class to whichequivalent receptors can gain access, and it is thisclass that is manifested by the AM pattern in theEEG. The formation of an AM pattern is by threeoperations requiring multiple sniffs for learning.One is habituation by which bulbar responses tounwanted input on non-reinforced trials are sup-pressed. The second is Hebbian learning by whichthe synapses between pairs of co-exited neuronsare strengthened [8, 10, 11]. The third step isread-out from the bulb to the prepyriform cortex.The pathway over which this is done is unlike theinput pathway to the bulb, which like all othersensory pathways has a topographic projection.The output pathway is divergent-convergent,meaning that each bulbar neuron transmitsbroadly, and conversely each receiving corticalneuron receives simultaneously from widely dis-tributed neurons in the bulb. The only bulbaractivity that is enhanced by the resulting spa-tiotemporal integral transformation of the bulbaroutput is that which has a common instantaneousfrequency, and that frequency is the carrier wave


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of the AM pattern formed by cooperative interac-tion. The sensory-driven microscopic activity is nottemporally coherent, so it is attenuated. As theresult, what the brain receives from the bulb is

what the bulb has constructed within itself, not thespecific information that has been imposed byreceptors during a sniff.

Together these three operations (habituation,association, and spatiotemporal integration)provide the abstraction and generalization that isnecessary for the acquisition of knowledge in thesense prescribed by Aquinas. It is noteworthy herethat the binding of neural activity is not that offeature detector neurons in discrete networks per-forming logical operations by synchronization offiring to represent an object [25, 42]. It is thecreation of a new AM pattern of activity thatembodies cumulative past experience with an odor-

ant upon presentation of that odorant, which en-ables the animal to modify itself in adding to itsexperience and to prepare itself for the next appro-priate action. As noted also by Hardcastle [27] and

long before by Aquinas [1], there is no basis forrepresentation of individual stimuli.

2.2. E6idence from neocortical EEGs

concerning reafference

The existence of comparable AM patterns hasnow been demonstrated in the visual, auditory,and somesthetic cortices [2]. The EEGs from ar-rays of electrodes on the pial surfaces show acommon aperiodic wave form on all electrodes,with a broad 1/f2 spectrum and with spatial AMpatterns that change with learning, which canserve as the basis for retrospective classification oftrials under discriminative conditioning. The AMpatterns are discrete events lasting 80120 ms andrecurring also at rates in the theta range, likeframes in a motion picture. The same lack of

invariance with respect to conditioned stimuli isfound with learning new stimuli in the appropriatemodality, and with contingency reversal. Eachlearned stimulus serves to elicit the construction ofa pattern that is shaped by the synaptic modifica-tions between cortical neurons from prior learning,which vastly outnumber the synapses formed byincoming sensory axons, and also by the brainstem nuclei that bathe the forebrain in neuromod-ulatory chemicals. Each cortical activity pattern isa dynamic operator that creates and carries themeanings of stimuli for the recipient animal. Itreflects the individual history, present context, andexpectancy, corresponding to the wholeness of the

intentionality. The patterns created in each cortexare unique to each animal. All sensory corticestransmit their output in the form of patterns ofmicroscopic action potentials into the limbic sys-tem, where they are combined to form gestalts.

Here it is important to review the architecture ofperceptual dynamics (figure 2). The mesoscopicpatterns that in mammals converge to the entorhi-nal cortex, prior to delivery into the hippocampusfor integration over time, all have the same for-mat. This indicates that the several AM patternscarried by action potentials can be as easily inte-grated with each other as was the case with themicroscopic sensory input to each primary cortex

by sensory action potentials, leading to the forma-tion of its mesoscopic pattern. The combined en-torhinal signal is sent from the outer three layersmainly by the perforant path to the hippocampus,and it is the main source of hippocampal input.The main target of hippocampal output in mam-mals is back to the inner three layers of theentorhinal cortex. Then the main target of trans-mission of the resultant integrated activity patterns

Figure 5. The sigmoid curve is shown for two values of Qm=1.82 and 5.0, with normalized pulse density, q, as a function ofnormalized wave density, 6, where Qm is the upper asymptoteof q. The derivative of the function, dq/d6, gives the nonlineargain. The peak gain is at the wave value, 6max (from [18]).

q=exp[(e61)/Qm], (1a)

dp/d6=exp[6(e61)/Qm], (1b)

6max=ln(Qm). (1c)


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is back to the primary sensory cortices. By thisarchitecture whatever signal that is provided byeach sensory port is combined and shared byre-direction back to the incoming ports. This re-

cursive architecture and process has been calledreafference [12, 50, 57], corollary discharge [46],and preafference [33]. Evidence for corollary dis-charges has been found in simultaneous recordingsfrom the bulb, prepyriform, entorhinal cortex, anddentate fascia in the hippocampus [34], showingthat prior to the arrival of an expected volley froman odorant, the olfactory bulb is prepared by brief,staccato signals from the entorhinal cortex toprime it for that stimulus.

This process was discovered by Helmholtz [28]in studies of patients with injury to the oculomotornerve, who reported that when they attempted tomove their eyes laterally, their visual field ap-

peared to move in the opposite direction. He alsoobserved that a patient who moved his eyeballpassively reported that the world appeared tomove, which it did not if the subject moved his eyeintentionally. He inferred that intentional eyemovement was accompanied by a discharge fromthe motor centers into the visual cortex, whichprovided the basis for distinguishing movement ofobjects in the visual field from apparent motionattributable to a movement of eyes or the head. Heconcluded that an impulse of the will that ac-companied intentional behavior was unmasked bythe paralysis. J. Hughlings Jackson [30] repeatedthe observations, but postulated alternatively that

the phenomenon was caused by an in-going cur-rent, which was a signal from the non-paralyzedeye that moved too far in the attempt to fixate anobject, and which was not a recursive signal froma motor centre. He was joined in this interpreta-tion by William James [31] and Edward Titchener[50], thus delaying deployment of the concepts ofneural feedback in re-entrant cognitive processesuntil late in the 20th century.

3. Characteristics ofmesoscopic dynamic brain states

The state of the brain is a description of whatit is doing in some specified time period. A statetransition occurs when the brain changes and doessomething else. For example, locomotion is a state,within which walking is a rhythmic pattern ofactivity that involves large parts of the brain,spinal cord, muscles and bones [43]. The entireneuromuscular system changes almost instantly

with the transition to a pattern of jogging orrunning. Similarly, a sleeping state can be taken asa whole, or divided into a sequence of slow waveand REM stages. Transit to a waking state can

occur in a fraction of a second, whereby the entirebrain and body shift gears, so to speak. The stateof a neuron can be described as active and firing oras silent, with sudden changes in the firing mani-festing state transitions. Populations of neuronsalso have a range of states, which are revealed bytheir characteristic wave forms, such as in slowwave sleep, fast activity in REM, epileptiformspikes in seizures, or the silence of brain death.The mathematics of nonlinear dynamics is de-signed to study these states and the transitions bywhich they are accessed and abandoned [10, 11,13, 14, 26].

3.1

. E6

aluation of the stability and destabilizationof mesoscopic states

A critical question to ask about the state of asystem is its degree of stability or resistance tochange. Evaluation is done by perturbing it andobserving over what range of intensity of perturba-tion it returns to its initial state [9, 11]. Forexample, an object like an egg on a flat surface isunstable, but a coffee mug is stable. A personstanding on a moving bus and holding on to arailing is stable, but someone walking in the aisle isnot. If a person regains his chosen posture aftereach perturbation, no matter in which direction

the displacement occurred, that state is regarded asstable, and it is said to be governed by an attrac-tor. This is a metaphor to say that the system goes(is attracted) to the state through an interim stateof transience. The range of displacement fromwhich recovery can occur defines the basin ofattraction, in analogy to a ball rolling to thebottom of a bowl. If the perturbation is so strongthat it causes concussion or a broken leg, and theperson cannot stand up again, then the system hasbeen placed outside its prior basin of attraction,and a new state supervenes with its own attractorand basin.

Stability is always relative to the time duration

of observation and the criteria for what is chosento be observed. In the perspective of a lifetime,brains appear to be highly stable, in their numbersof neurons, their architectures and major patternsof connection, and in the patterns of behavior theyproduce, including the character and identity ofthe individual that can be recognized and followedfor many years. Brains undergo repeated transi-tions from waking to sleeping and back again,


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coming up refreshed with a good night or irritablewith insomnia, but still, giving arguably the samepersons as the night before. Personal identity isusually quite stable. But in the perspective of the

short term, brains are highly unstable. Thoughtsgo fleeting through awareness, and the face andbody twitch with the passing of emotions.Glimpses of their internal states of neural activityreveal patterns that are more like hurricanes thanthe orderly march of symbols in a computer. Brainstates and the states of populations of neuronsthat interact to give brain function, are highlyirregular in spatial form and time course. Theyemerge, persist for a small fraction of a second,then disappear and are replaced by other states.Yet by the criterion of the statistical parameters ofthe electrical pulse and EEG activities of its popu-lations of neurons, the waking cerebral cortex is

remarkably robust. The stability is readily demon-strated by perturbation with sensory and electricalafferent stimuli, which displace the cortex brieflyfrom its background level of activity, to which itreturns by relaxation in a trajectory that is mani-fested in event-related potentials. The range ofstimulation over which the return can take placedefines the basin of attraction for the backgroundattractor. Parameters such as the intensity atwhich there is failure to return owing to the onsetof epileptic seizure, or the duration at which exces-sive repetition induces sleep, define the boundaryor separatrix of the waking basin from otherbasins in a landscape of cortical basins of

attraction.

3.2. Three types of stable cortical states

In using dynamics we define several kinds ofstable state, each with its type of attractor. Thesimplest is the point attractor. The system is at restin an unchanging steady state unless perturbed,and it returns to rest when it is allowed to do so.As it relaxes to rest as when an area of cortexgenerates an evoked potential, it has the history ofwhat happened, but that history is lost after con-vergence to rest. Examples of point attractors aresilent neurons or populations that have been iso-lated from the rest of the brain, and also the brainthat is depressed into inactivity by injury or bydeep anesthesia, to the point where the EEG hasgone flat (figure 6, bottom trace; figure 7, verticalline at the base; figure 8, lowest frame).

A special case of a point attractor is noise. Thisnoise is observed at the microscopic level in popu-lations of neurons in the cortex that are coupled

Figure 6. Four levels of function of the olfactory system arerevealed by EEG recording. The lowest is the non-interactiveopen loop state [14] imposed by deep anesthesia, which sup-

presses brain activity. The next is the awake state of rest withaperiodic waves giving a 1/f2 spectrum. The level in whichbehavior is generated reveals recurrent state transitions withinhalation, by which bursts are formed that reveal spatialpatterns of AM (amplitude modulation) relating to odorantrecognition. The upper trace shows the pattern of high-ampli-tude spikes sustained at 3/s when an epileptic seizure has beentriggered by intense electrical stimulation. This state is likewisechaotic, but with a reduced correlation dimension (from [45]).

by synaptic excitatory feedback, without need forinhibition, because their interactive state is stabi-lized by the refractory periods of neurons in popu-lations [9, 11, 13]. The neurons fire continually atirregular times, so that their autocorrelation func-tions go rapidly to zero, and their interval his-tograms decay exponentially after a dead timeimposed by the absolute and relative refractoryperiods of the action potentials. Their crosscorrela-tion functions likewise are near zero, because theydo not fire in concert with each other. Knowledgeabout the prior pulse trains from each neuron andthose of its neighbors up to the present fails tosupport prediction of when the next pulse willoccur. The state of noise has continual activitywith no history of how it started, and it gives onlythe expectation that its mean and higher orderstatistical properties will persist as in the recentpast. Simulation of the stable and sustained outputof a population of mutually excitatory neurons isshown in figure 9.

An amplitude histogram of the pulse density ofneurons in local neighborhoods at the mesoscopiclevel shows a Gaussian distribution about a meanvalue (figure 10), which is restricted to the upperregion of the nonlinear sigmoid gain curve (figure5) that specifies the relation between the dendriticcurrent density, 6, and the axonal pulse density, q,


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of neural populations in normalized coordinates[14]. The derivative of the sigmoid curve revealsthe nonlinear gain, dq/d6. In the upper region,6\6max, an excitatory perturbation increases the

excitatory activity of the population and decreasesthe gain, owing to the cumulative effects of therefractory periods, and when the gain goes down,the level of evoked activity decreases until theactivity has returned to the self-sustained activitylevel. Any decrease in activity is accompanied byan increase in gain, which is followed by a com-pensatory homeostatic increase in activity. Thusthe stability of the self-sustaining state is revealedby perturbation [14] to have the form of a pointattractor. At the microscopic level, the self-sus-tained activity appears densely packed action po-tentials (noise), and at the mesoscopic level itappears as an excitatory bias (figure 9, top trace).

The power spectral density has the 1/f2

form(figure 10, upper frame) that corresponds to thespectrum of brown noise created by white noisepassed through a two-pole low-pass filter).

The interaction of excitatory neuron popula-tions with inhibitory populations at the meso-scopic level gives sustained oscillation. In someconditions imposed by surgical or pharmacologicalisolation of the olfactory bulb [14], the oscillationmay take the form of a sinusoidal wave [9, 18] that

Figure 8. This perspective offers a hypothesis of how an attrac-tor landscape of learned basins of attraction is created witheach inhalation. The selection of a basin is made by the inputodorant. If the stimulus is novel or unknown, an orientingbehavior occurs (I dont know), and the system goes into thesurrounding chaotic basin, which provides the aperiodic unpat-terned activity that drives Hebbian learning for formation of anew basin. Attractor crowding [59] then changes all of the otherbasins, resulting in contextual pattern dependence (figure 4)(from [45]).

Figure 7. The four states of the olfactory system as revealed byEEGs are linked in a bifurcation diagram, in which the ampli-tude of olfactory neural activity controlled by brainstem mech-anisms serving arousal is used as a bifurcation parameter (from[45]).

gives a peak at one frequency in the power spectraldensity. An open system that gives periodic behav-ior is said to be governed by a limit cycle attractor.The classic example is a wrist watch. If it keepsgood time despite being knocked about, it is sta-ble, but if it winds down and runs out of power, itbecomes unstable and switches to a point attrac-tor. Its history is then limited to one cycle, afterwhich there is no retention of its transient ap-proach in its basin to its attractor. Neurons andpopulations do not usually fire periodically, andwhen they appear to do so, close inspection showsthat the activities are in fact somewhat irregularand unpredictable in detail. When nearly periodicactivity does occur, at the microscopic level it isdue to injury or pacemaker potentials. At themesoscopic level it is seen in the olfactory bulbwhen it has been surgically or cryogenically iso-lated from the rest of the system [14]. At themacroscopic level of behavior, periodicity is eitherintentional, as in rhythmic drumming, clapping


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and dancing, or it is pathological, as in the peri-odic oscillations of the eyes in nystagmus or of thelimbs during Parkinsonian tremor or of the cortexduring the hypersynchrony of partial complex

seizures that are revealed by near-periodic spiketrains (figure 6, top trace). The reason that limitcycle oscillation is uncommon at the mesoscopiclevel is that the feedback is nonlinear. The wavedensity and pulse densities show Gaussian distri-butions (figure 11) about a mean that is located onthe low amplitude segment of the sigmoid func-tion, below the peak of the nonlinear gain (figure5). When activity levels increase, the gain in-creases, causing a regenerative positive feedbackthat increases activity still further. This nonlinear-ity is the basis for the nonlinear state transitionsthat recur with respiration in the olfactory system(figure 3). Figure 10. The power spectrum and amplitude histogram are

from the pulse density output of an element simulating a localneighborhood in the periglomerular population in the outerlayer of the olfactory bulb (from [18]).

Figure 9. Representative state variables in a 2 s time period areshown from a set of coupled nonlinear ordinary differentialequations (ODEs) solved on a digital computer. Random num-ber generators were used to simulate the noise to stabilizechaotic dynamics. The EEGs and pulse densities observed inthe various parts of the olfactory system were simulated. Al-though the several parts are continuously interacting to gener-ate the aperiodic time series, the crosscorrelations between thetraces average near zero, illustrating the failure of techniques oflinear correlation to detect nonlinear interactions (from [18]).

The third type of attractor gives aperiodic oscil-lation of the kind that is observed in recordings ofEEGs. There is no one or small number of fre-quencies at which the system oscillates. The systembehavior is therefore unpredictable, because per-formance can be projected well into the futureonly for periodic behavior. The existence of aperi-odic oscillation, now widely known as chaotic,was known a century ago, but systematic studywas possible only after the full development of

digital computers. The best known systems with

Figure 11. The power spectrum and amplitude histogram arefrom the pulse density output of an element simulating a localneighborhood in the mitral cell population, the projectionneurons of the olfactory bulb (from [18]).


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chaotic attractors have a small number of compo-nents, finite fractal dimension, and few degrees offreedom, as for example, the double-hinged pendu-lum, the dripping faucet, and the Lorenz, Chua,

and Rossler attractors. These simple models arestationary, autonomous, and noise-free, belongingto the class of deterministic chaos.

Large and complex real-world systems, whichinclude neurons and neural populations are noisy,infinite-dimensional, non-stationary, non-au-tonomous, yet they are capable of rapid statetransitions, indicating behavior that is distin-guished as stochastic chaos [15]. The source inthe olfactory bulb has been shown to be the synap-tic interaction of millions of mutually excitatoryneurons, which create a 2-D field of microscopicbrown noise in the bulb, but which are constrainedby their relative and absolute refractory periods to

generate a mesoscopic order parameter (a steady-state excitatory bias under brainstem control) thatregulates the threshold for state transitions to formthe spatiotemporal AM patterns of cortical activ-ity revealed by the EEG. The bias provides thebifurcation parameter that takes the olfactory sys-tem from a point attractor under deep anesthesia(figures 6, 7) to a chaotic attractor in waking rest,and then to the level of arousal at which thevolleys of action potentials from receptors in thenose during inhalation can readily change thelandscape and induce a state transition to one ofthe newly actualized basins of attraction (figure 8,inhalation).

EEG recordings from epidural electrode arraysfixed permanently onto the visual, auditory, andsomesthetic cortices have shown that each sensorycortex (comparable in this respect to the olfactorybulb, the first cortical stage in smell) maintainsmultiple basins corresponding to previouslylearned classes of stimuli, as well as to the unstim-ulated state, which together form an attractorlandscape. The postulate here (figure 2) is thatpreafferent input [33, 34] from the limbic systemcan serve to bias the landscape in such a way as tofacilitate the capture of the system by a basin of anattractor (figure 8) corresponding to the goal ofthe intended observation, perhaps in the mannerof the variable tiling in a Voronoi diagram.Thereby basins of attraction in each of the sensorycortices are shaped by limbic input to sensitize thereception of the desired class of stimuli in everymodality. These chaotic prestimulus states of ex-pectancy establish the sensitivities of the cortices,so that the very small number of sensory actionpotentials evoked by expected stimuli can carry the

cortical trajectories into the basins of appropriateattractors in each modality, irrespective of whichequivalent receptors actually receive the expectedstimuli. In the absence of the stimulus, the cortices

continue to transmit their output to the limbicsystem, confirming the continuing absence. Thepreafference is facilitated by the motor systemsthrough orientation of the sensory receptors inspace by sniffing, looking and listening.

4. Understanding brain chaos by modelingwith differential equations

The discovery of chaos has profound implica-tions for the study of brain function [45]. Achaotic system has the capacity to create novel and

unexpected patterns of activity. It can jump in avirtual instant from one mode of behavior toanother, because it has a collection of attractors,each with its basin, and it can move from one toanother in an itinerant trajectory [53]. It retains inits pathway across its basins its history, whichfades into its past, just as its predictability into itsfuture decreases. Transitions between chaoticstates constitute the dynamics that we need modelin order to understand how brains do what theydo.

4.1. The importance of noise for stabilizationof brain chaos

Systems that model neurons and brains withnetworks of nonlinear ordinary differential equa-tions (ODEs) have multiple point, limit cycle andchaotic attractors, each with its basin of attraction,which serves to provide the generalization gradientrequired for classification of recurring stimuli thatare never twice the same. If the basin is that of apoint or a limit cycle attractor, the system canproceed to a predictable and identical end state. Ifthe basin leads to a chaotic attractor, the systemgoes into ceaseless fluctuation, as long as its energylasts. If the starting point is identical on repeatedtrials, which can only be assured by simulation ofthe dynamics on a digital computer, the sameaperiodic behavior appears. If the starting point ischanged by an arbitrarily small amount, althoughthe system is still in the same basin, the trajectoryis not identical. A deterministic chaotic system thatis in the basin of one of its chaotic attractors islegendary for its sensitivity to the initial condi-tions. If it is not known to an observer whether


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there is a difference in starting conditions, theobserver can infer the difference from the unfold-ing behavior of the system, if a difference in tra-

jectories becomes apparent. This outcome shows

that a chaotic system has the capacity to createinformation in the course of continually construct-ing its own trajectory into the future, but that theinformation exists not in the system but in therelation of the observer to the system. That infor-mation is not useful for pattern classification inthe circumstance that every new stimulus of thesame class is different in irrelevant detail from anypreceding, so that while the deterministic chaoticsystem provides extreme sensitivity, it does notreadily provide for generalization.

Stochastic chaotic systems feed on noise, whichmakes them robust in respect to initial conditionsand therefore suitable for pattern recognition.

Digital simulation of stochastic chaotic dynamicspresents special problems, due to the discretizationthat is inherent in using rational numbers insteadof real numbers to simulate continuous variablesin time, space and intensity. Large systems using\100 ODEs suffer from an explosion of attrac-tors of all types, leading to the phenomenon ofattractor crowding [59], in which the size of thebasins shrinks until it approximates the step sizein digital simulation. Tracking the true shadow-ing trajectory [22, 24] by discrete steps in numeri-cal integration, like trying to walk a straight linesuch as a curbstone while intoxicated. When thetrue trajectory comes close to the edge of a

chaotic basin with a fractal boundary, measuredin terms of the number of bits in digital approxi-mation of the continuum, the next discrete steptakes the system into the adjacent basin, which ismost likely to be that of a point or limit cycleattractor, and which will lock the system into aperseverative computational loop. With elapsedtime this numerical failure becomes virtually cer-tain. For example, in simulation of EEGs with theKIII model [14] incorporating 920 ODEs thatare solved using double precision arithmetic givinga precision of 1 part in 1016, the model escapesfrom its designated chaotic basin in a simulatedrun time of 1.5 s, and it must then be shutdown and restarted.

In the KIII model of the olfactory system thisproblem of numerical instability has been solvedby the introduction of a random number genera-tor (which is actually a deterministic chaotic gen-erator that is capable of demonstrating thesensitivity to initial conditions) to simulate addi-tive noise modeled on noise sources in the olfac-

tory system. Two types of noise are used at a levelof 515% of the amplitude of the state variables:full wave rectified noise independently generatedat each input to simulate the excitatory receptor

pulse densities, and Gaussian noise to the elementcorresponding to the anterior olfactory nucleus tosimulate the sum of centrifugal input to the olfac-tory system constituting spatially coherent noise[18]. The additive noises suffice to stabilize thedigitally embodied attractors [32, 38], and theyenhance the capabilities of the KIII model forpattern classification. This is analogous to the useof noise to improve system performance instochastic resonance [5] and in simulated anneal-ing [38], but those uses serve to destabilize systemsthat are captured by local minima, whereas thenoise in the KIII model serves to expand thebasins of the learned attractors, which otherwise

are excessively narrow.Examples are shown (figure 9) of simulated ac-

tivity densities from the olfactory bulb (the peri-glomerular cells and the mitral cells), the anteriorolfactory nucleus (pyramidal cells) and theprepyriform cortex (the superficial pyramidalcells), which generate the EEGs. The transientstage of initialization takes about 250 ms. Anexample is shown in figure 10of the power spectraldensity versus frequency in Hz in log-log coordi-nates for the periglomerular cell pulse density as afunction of time (upper frame) after the steadystate has been reached. The amplitude histogram isshown in the lower frame, superimposed on the

relevant portion of the nonlinear gain curve (figure5) at the appropriate value for the maximalasymptote, Qm=1.7. As explained in text, thelocation of the distribution of amplitudes of pulsedensity above the maximal gain, here at the natu-ral logarithm of wave amplitude of 0.53, is pre-dicted from the fact that the periglomerular cellpopulation is governed by a non-zero point attrac-tor (the zero point attractor being displayed underdeep anesthesia (figure 6)). The example of thepower spectral density (figure 11) reveals in thegamma range (20 80 Hz) the excess of powerabove a linear 1/f2 relation that is characteristicof olfactory system EEGs. The amplitude densitiesare distributed about a mean wave amplitude nearzero, and below the peak of the gain curve at waveamplitude of 1.6 (natural logarithm of the maxi-mal asymptote, Qm=5.0 (figure 5)). The locationof this distribution is consistent with the gover-nance of the mixed population of excitatory andinhibitory bulbar neurons by a limit cycle attrac-tor. However, periodic oscillation in the bulbar


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EEG is only seen when the bulb has been isolatedfrom the rest of the olfactory system. The anteriornucleus and prepyriform cortex also have charac-teristic frequencies that are measured as in the

bulb from averaged evoked potentials [14], but thethree frequencies are incommensurate (not in in-teger ratios), so when they are coupled by feed-back into a neural menage a trois, and chaoticoscillation results.

4.2. Measuring phase by fitting wa6eletswith nonlinear regression to EEGs

The aperiodic carrier wave form of each spatialAM pattern has a dominant or center frequencyof the oscillation, at which the phase can be deter-mined with respect to the phase of the spatialensemble average. Measurements of phase weremade to test three predictions. The first was thatphase gradients would be detected in spatial pat-terns of gamma bursts, which would correspondto the direction and velocity of the afferent path-ways, as had already been demonstrated in thespatial patterns of oscillatory evoked potentials onelectrical stimulation of the afferent axons to theolfactory bulb and to the prepyriform cortex [9].The second prediction was that the phase dis-tribution in olfactory AM patterns would be con-verged (narrowed) during a sniff of a familiarodorant, indicating an approach of the distribu-tion of oscillations to zero time lag [35, 41],which was based on the binding hypothesis [25,42]. The third prediction was that the width of thedistributions of phase within AM patterns wouldbe limited to a range in which those patternscould be extracted by spatial integration withoutdegradation by excessive phase lag (which wouldoccur beyond 945, at which the shared power ishalf the total).

EEG traces were measured by decomposingthem with appropriate basis functions. In theFourier case, the basis functions were cosines giv-ing the phase and amplitude of frequencies in thegamma range. Whereas accurate measurement ofamplitude was relatively easy, measurement of thephase was subject to large errors owing to the briefduration of segments, the small number of cyclesin each segment, the strong tendency to frequencymodulation (FM) and amplitude modulation(AM) about center frequencies and amplitudes inboth time and space, and the mix of multiplefrequencies in spectra of most segments, especiallyfrom neocortices with their tendency to 1/f2 spec-tra [2]. An alternative was to use wavelets, which

were cosines linearly modulated in amplitude(AM) and frequency (FM) in time about centervalues [21]. The sum of two wavelets was fitted tothe 64 EEG traces in concomitantly recorded seg-

ments lasting 64128 ms by nonlinear regression,using the least mean square residuals as the crite-rion of convergence, each wavelet being given by:

EEG(~)=

%2

n=1

6n,i [1+AMi](~~m)

cos{2yfn[FMi(~~m)]~+n,i}+m(~),

i=1, 64 (2)

where n=1, 2 was the number of the wavelet, 6n,iwas mean amplitude of the i-th component, AMiand FM

iwere fixed coefficients to represent linear

amplitude and frequency modulation across thecenter of the time segment, ~m, fn was the commoncenter frequency of wavelet oscillation, n,i was thedesired phase of onset of the i-th damped cosine,and m(~) was the residual noise after fitting the sumof two cosines to the EEG(~) segment. The sumwas fitted in stages to the spatial ensemble averageEEG in order to estimate the frequency, modula-tion coefficients, and the reference phase,

n, with

respect to which all other 64 phase values, n,i, andmean amplitudes, 6n,i(~), were estimated by fittingthe same sum of two component cosines to each ofthe 64 traces, i=1, . . . , 64.

The standard error of measurement (SEM) wasestimated by measuring a single cosine embeddedin noise in 64 traces at zero phase and varyingamplitudes corresponding to observed EEG ampli-tude distributions in AM patterns, with presetlevels of random numbers filtered in the same wayas the EEGs to give specified signal:noise ratios[16, 21]. The results gave a SEM that averagedabout 960 for a single cosine fitted to rawEEGs. The SEM was reduced to 96 after appli-cation of appropriately designed wavelets spatialand temporal pre-filters, which have since beenused in all studies requiring the phase measure-ments presented here.

A major problem in correlation of intentionalbehavior with neocortical EEGs (as comparedwith olfactory EEGs) was temporal segmentation.The location of spatial AM patterns in the gammarange of olfactory EEGs was simplified by theprominent respiratory wave in the theta range andthe associated temporal AM revealing bursts ofgamma oscillations. Contrariwise, visual inspec-tion of the neocortical EEGs gave no indication of


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Figure 12. Upper frame: % The time series of residuals is from repeatedly fitting equation (2) by nonlinear regression to EEGsegments lasting 62 ms (32 time points at 2 ms intervals). Time is in 2-ms bins after stimulus arrival at 0. Lower Frame: % Residualsfrom nonlinear regression of a cone onto the 64 values of phase from the dominant wavelet (n=1 in equation (2)) fitted to the 64EEG traces at each frame step. The reference phase was that of the spatial ensemble average. Acceptable fit for cone detection wasB20% residuals.

where AM pattern segments might start or end,either by any distinctive wave in the theta andalpha ranges, or by any temporal AM in thegamma range. Therefore, a fixed-duration segmentwas stepped along the multiple EEG traces onevery trial. The measurement procedure was re-peated on 32-bin (64 ms) segments stepped at 2 bin

(4 ms) overlapping intervals over time. The good-ness of fit of the two cosines to the 64 EEG traceswas shown by plots of residuals. An example ofthe results from a single trial is shown in figure 12,upper frame, where residuals B20% was adoptedas a criterion of successful fit.

4.3. Testing predictions of propertiesof spatial phase distributions

The first prediction, that gamma phase gradientswould correspond in direction and slope to thedirection and conduction velocity of afferent ax-ons, was confirmed for the prepyriform cortex butnot for the olfactory bulb, despite the fact that thespatial phase gradients of the bulbar oscillationsevoked by single-shock electrical stimulation of theprimary olfactory nerve conformed to the proper-ties of its axons [9]. About 50% of the variance ofthe 64 phase values was captured by a plane, butthe orientation of tilt of the plane varied randomlyfrom each AM pattern to the next, and without

relation to the conditioned stimuli used for testingbehavioral discrimination. The second prediction,that the phase distribution would narrow or col-lapse on arrival of an expected conditioned stimu-lus and formation of an AM pattern signalingbinding, failed in both the bulb and the prepyri-form cortex. No differences in the standard devia-

tions of phase distributions were found betweencontrol and test periods. The third prediction, thatthe distributions of phase values would be lessthan 945, was confirmed within the window ofthe arrays (44 t o 66 mm), but the entiresurfaces of the structures being recorded fromwere inaccessible for direct observation.

Visual inspection of 2-D contour plots of bulbarnumerical phase values not uncommonly revealedradial symmetry in the form of concentric isophasecontours. With this clue the phase distributions ofAM patterns were fitted using nonlinear regressionwith a cone having the parameters of location in2-D of the apex and the slope of the cone, includ-ing its sign. The goodness of fit improved substan-tially over fitting a plane to the phase data. Theresults from three sessions of twenty trials in eachof five rabbits are shown in figure 13, in which thesize and location of the array was superimposedon an outline of the olfactory bulb, the bulb wasapproximated by a sphere, and the bulbar surfacewas displayed as a flattened sphere [16]. The apices


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Figure 13. Phase distributions were measured with respect tothe phase of the spatial ensemble average at the surface of theolfactory bulb and fitted with a cone in spherical coordinates.The sketch is a projection of the outline of the bulb as it wouldappear on looking through the left bulb onto the array an thelateral surface of the bulb. A representative set of isophasecontours is at intervals of 0.25 radians mm1. The locationsof the apices of the cones on the surface of the sphere (2.5 mmin radius) are plotted from the center of the array to theantipode. The square outlines the electrode array. The standarderror of location of points was twice the radius of the dots(from [16]).

the amplitude peaks or valleys in the accompany-ing AM patterns. The signs of the apices (leadindicated by a solid dot and lag by an open dot)varied at random on sequential trials. The slopes

of the phase cones measured in radians/mm variedwith gamma frequency in Hz, but when convertedt o m s1 using the frequency converged to aninvariant at 1.82 m s1, which was the estimatedfrequency of axon collaterals of mitral and tuftedcells running in the bulb parallel to the surface.Considering the distance around the bulb at thelevel of the internal plexiform layer, the estimatedmaximal phase difference between any two pointsin the mitral cell layer was estimated to be lessthan 945 over the full gamma range (2080 Hz).

Comparable spatial phase cones have beenfound in conjunction with AM patterns in visual,auditory, and somesthetic EEGs, both by Fourier

decomposition [17] and by use of wavelets (equa-tion (2)) and nonlinear regression. A typical se-quence of goodness of fit in terms of % residualsfrom the post-stimulus segment of a single trial isseen in figure 12, lower frame, where B20% resid-uals was adopted as one of several criteria fordetermining the presence of a cone. A raster plotin figure 14 shows an example of the time locationsof recurring phase cones in a set of forty trials.Figure 15 shows the spectrum of the autocovari-ance of the cones in the forty trials laid consecu-tively in comparison with the autospectrum ofautocovariance of the concomitant spatial ensem-ble average of the EEG and the cospectrum. In

every rabbit and session a prominent peak wasfound in the theta range (27 Hz) of the cospec-trum of the phase cones and the EEG. A peak inpower was found often but not always at the same

were extrapolated to locations anywhere on thesurface, but not to the posterior quadrant wherethe connections to the forebrain are located, andwithout relation to the stimulus conditions or to

Figure 14. Distribution are shown by line segments of the times of occurrence of phase cones during forty trials by one rabbit in onesession. Data are from visual cortex; the cones were fitted in planar coordinates. Each trial lasted 6 s (the display is truncated at 400ms and 5 600 ms by temporal filtering), with the stimulus arriving at the middle (3 000 ms). The phase cones occur equally often inthe control and test periods, as expected in self-organizing cortical dynamics.


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Figure 15. The spectrum of the autocovariance of the cones inthe forty trials laid consecutively is compared with the au-tospectrum of the autocovariance of the concomitant spatialensemble average of the EEG and the cospectrum. The peak inthe theta range of the cospectrum is an indication that adestabilizing forcing function originating extracortically oper-ates on the sensory neocortices in a manner similar to the role

of respiration in olfaction [14].

the 64 electrodes cannot be ascribed either toactivity at the reference electrode on monopolarrecording or to volume conduction from a singlecurrent generator under the array, because of the

systematic differences in phase. Nor is it due todecorrelation with distance over a distribution ofnoise generators smeared by volume conduction[6], because that would not give radial phase gradi-ents. Nor does it allow for progressive entrainmentof coupled oscillators, because the phase gradientspersist through the segments, indicating that aphase gradient established in the time of transitionlasting a few ms persists through the 80120 msduration of the AM pattern, probably owing to thesparseness of the local connection densities [4, 17],which fail to support convergence of activity intozero time lag synchrony within the duration of thepatterns. These results, however, give support to

recent findings of the spatial coherence of gammaactivity recorded on the scalp of humans perform-ing cognitive tasks [48, 49], because the size of theAM patterns in cm may allow detection of thegamma bursts in scalp EEGs.

Second, the random variation of the sign of theapices cannot be explained by an intracortical orthalamic pacemaker, which could only have phaselead whether acting by excitation or inhibition. It iscompatible with a symmetry-breaking state tran-sition such as a saddle node bifurcation. The prop-erty of mesoscopic states that makes theminteresting is the capacity they give to large popula-tions of neurons for rapid changes in the global

spatiotemporal patterns of organization and func-tion. Some well known examples are the transi-tions between waking and sleeping states, betweenvocalizing and swallowing, and between walkingand running, by which the neurons widely dis-tributed in the brain and spinal cord shift theirfiring from one coordinated pattern to the nextglobally in a few ms. Evidence from physical dis-tributed systems shows that state transitions donot start simultaneously throughout the systemsbut begin at a site of nucleation and spreadradially, as in the formation of a snowflake orraindrop around a dust particle. The radial phasepatterns in the EEG provide strong evidence thatAM patterns form by self-organizing cortical statetransitions. They indicate that the cortical activitymanifested in the EEGs is not shaped directly bysensory or other external sources of pulses, butinstead that it is endogenously constructed follow-ing state transitions that direct the corticesthrough attractor landscapes yielding differingspatial patterns [1116]. In the olfactory system it

frequency in the EEG autospectrum, but a peakwas seen rarely in the autospectrum of the phasecones.

As in the olfactory bulb the locations and signsof the apices of the phase cones varied at randomover successive events, and the frequencies of thegamma oscillations varied also. Typical distribu-tions are shown in figure 16 of the center frequen-cies in Hz and the slopes in radians mm1. Fromthese were calculated the phase velocities in m s1

(which are consistent with reports of the distribu-tions of conduction velocities of neocortical axons[47, 17]), and the radii to the phase contour at945, from which the half-power diameters indi-cated by the phase cones were calculated.

4.4. Radial phase gradients pro6ide physiologicale6idence for first order cortical state transitions

Several inferences follow from these findings.First, the common wave form found in traces from


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is apparent that the state transitions depend onreceptor input, which provides a behaviorally re-lated forcing function that can destabilize thebulb. If this hypothesis is correct, then the forma-

tion of AM patterns and phase cones in neocortexrequires comparable forcing functions. Evidencefor these is not readily seen in EEGs, but seemsunequivocal in cospectra of EEGs with sequencesof phase cones. Further evidence for forcing func-tions should be sought in behaviors such as sac-cades in vision, finger tremor in touch, and

efferent control of muscles in the middle ear inaudition.

Third, the delays in axonal propagation mani-fested in the phase cones may serve to delimit soft

spatial boundaries of neocortical AM patternsduring both construction and read-out. Theneocortical neuropil forms a continuous sheet, asshown by the phenomenon of spreading depres-sion of Leao [3], which stops only at the borders ofthe neocortex in each hemisphere with the ar-chicortex and the callosum. The submillimeter mi-

Figure 16. Upper left: An example is shown of the distribution of center frequencies (Hz) of phase cones. Upper right: Thedistribution of slopes (radians mm1) of phase cones, was bimodal, showing equal probability of maximal lead or lag at apices.Lower left: The phase velocities of slopes calculated from the center frequencies were compatible with estimates of the conductionvelocities of axons running parallel to the pia, which are slower than so-called U-fibers running deeply from one area of cortex toanother. Lower right: The mode of the distribution of half-power diameters is about 1.1 cm, which is much larger than the sizes ofcortical columns and hypercolumns.


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croscopic architecture of cortex is spatially coarse-grained by its input projections into corticalcolumns and barrels, which are smaller by anorder of magnitude than the mesoscopic AM pat-

terns. The EEGs of cortical areas that are sepa-rated by macroscopic distances lack high spatialcoherence, giving evidence that the mesoscopicAM patterns must have soft boundaries. The obli-gatory axonal propagation delays may provide theneocortical dynamic boundary conditions, whichare required to give the different areas a requisitedegree of autonomy, while not freezing them intoanatomically fixed arrangements. Within eacharea, the spatially coherent EEGs manifest cooper-ative interactions among millions of neurons, butcooperativity must weaken with phase dispersion.The radial phase gradients can serve to attenuatemesoscopic synaptic interactions with distance.

With no hard edges the half-power radius (9cos45) can serve to define the sizes and locate the softboundaries for local cooperative domains, at theinterface between microscopic neural activity andcortical mesoscopic states. The cosine values thenprovide a measure of the degree of relatedness ofthe gamma activity within a temporal segment ofthe EEG in a mesoscopic cortical area.

Fourth, the classification of AM patterns withrespect to CSs reveals that the classificatory infor-mation is homogeneously distributed in space; nochannel is any more or less important than anyother [2, 16, 20]. This property shows that theinformation relating to the topographic mapping

of the sensory input has been spatially dissemi-nated by the dynamic operation of constructingAM patterns. The independence of the mesoscopicpattern from the details of the input-dependentcortical architecture may be critical for the integra-tion of multisensory percepts, in which the localspace-time gradients peculiar to the retinal, coch-lear and cutaneous mappings are no longerrelevant.

Fifth, phase cones are not detected in prepyri-form cortex, which leads to the suggestions thatthey are a property of self-organizing dynamics inprimary sensory areas of cortex, and that a sec-ondary area of cortex is required to be driven bythe AM pattern in order to transform it to a newpattern that is compatible with input requirementsof other parts of the brain. In other words, pairs ofcortices may work in tandem, one to produce theAM patterns, the other to prepare them for furthertransmission, and to read and interpret the outputof the first cortex. If so, then the phase cone servesas an identifying property of the first type in

sensory neocortex, and a search should be madefor cortices of the second type corresponding tothe role of the prepyriform cortex as a readoutsystem, while the anterior nucleus serves as a

controller of the olfactory chaotic dynamics [5].

5. Conclusions

The sensory cortices are conceived as beingbistable, having serial receiving and transmittingmodes that subserve feature binding during dias-tole and the formation and transmission of percep-tual constructs in AM wave packets [9] duringsystole forced by input. Each burst constitutes thesystolic output of a sensory area, which is pre-pared during a pre-burst diastolic period of sen-sory intake ending in a first order state transition,and which is constructed by the self-organizingattractor dynamics of the neural population ineach area. Whatever the clean logical functions ofcortical microscopic neural networks might be, thenonlinear dynamics of cortices at the mesoscopiclevel by this hypothesis are neither autonomous,stationary, or noise-free, but engaged with inputand output, undergoing repeated changes in state,and embedded in the noise of myriads of micro-scopic action potentials constituting mesoscopicbiases [13, 14].

These considerations lead to the inference thatthe wave packets are formed by cooperative synap-tic interactions within areas of cortex on a meso-scopic scale greater than the microscopic time andspace scales of the component neurons, but smallerthan the macroscopic scales of domains shown bybrain imaging based on metabolic and hemody-namic processes. These mesoscopic interactionsshape the spatial AM patterns owing to changes inintracortical synaptic strengths from cumulativeeffects of reinforcement learning. Each successivespatial AM pattern is formed by a nonlinear statetransition having an estimated state transition timeon the order of 6 ms and a modal diameter on theorder of 5 to 20 mm, considerably greater than thedimensions of the dendritic arbors of most neu-rons and the estimated sizes of cortical columnsand hypercolumns, but less than the dimensions ofthe lobes of the forebrain and, in small mammals,the cerebral hemispheres [17].

The significance of these findings and inferenceslies in the possibility that the spatial coherencesmanifest mesoscopic integrative processes bywhich information that is injected into corticalactivity sensory receptors through thalamic relaysis integrated into perceptual forms shaped by life--


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long prior learning. If so, the processes offer analternative to a widely entertained form of thebinding hypothesis [44, 56], which holds that thepulse trains of a small network of microscopic

neurons are locked in zero phase and zero time lagsynchrony [52, 54]. Instead, the binding revealed atthe mesoscopic level consists in a small covariantfraction of the total variance of all the neuronswithin the diameter of a wave packet imposed bycooperative dynamics, which provides fixed phasevalues of the oscillating probabilities of neuralfiring over its time duration, but which imposes adistribution of their phase values over its spatialextent.

These new data and inferences have profoundimplications for studies of cognitive brain func-tions in humans, because they open the way toexperimental investigations of human EEG activ-

ity both at the scalp and at the pial surface bygiving assurance that recording with high-densityelectrode arrays can yield information of greatvalue for understanding normal cognition and di-agnosing and treating cognitive disabilities [20],arising primarily from disorders at the mesoscopiclevel and secondarily manifesting themselves at themicroscopic level in biochemical abnormalities andat the macroscopic level in distressed behaviors.These findings also are important for engineers inthe design of new kinds of intelligent machines[14], in which the emphasis is placed on self-orga-nization and the emergence of intentional behaviorthat is created by nonlinear dynamics modeled on

the distributed, continuous-time dynamics of thecerebral cortex at the mesoscopic level.

Acknowledgements

This work was supported by grants from theNational Institute of Mental Health MH06686, theOffice of Naval Research N00014-90-J-4054, andfrom ARO MURI DAAH04-96-1-0341.

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mesopic neurodynamics - from neuron to brain

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