the labile brain. i. neuronal transients and nonlinear ...karl/the labile brain i.pdf · the labile...

The labile brain I Neuronal transientsand nonlinear coupling

Karl J FristonWellcome Department of Cognitive Neurology Institute of Neurology Queen Square London WC1N 3BG UK

(kfristoncentlionuclacuk)

In this the centrst of three papers the nature of and motivation for neuronal transients is described inrelation to characterizing brain dynamics This paper deals with some basic aspects of neuronaldynamics interactions coupling and implicit neuronal codes The second paper develops neuronaltransients and nonlinear coupling in the context of dynamic instability and complexity and suggests thatinstability or lability is necessary for adaptive self-organization The centnal paper addresses the role ofneuronal transients through information theory and the emergence of spatio-temporal receptive centeldsand functional specialization

By considering the brain as an ensemble of connected dynamic systems one can show that a sucurrencientdescription of neuronal dynamics comprises neuronal activity at a particular time and its recent historyThis history constitutes a neuronal transient As such transients represent a fundamental metric ofneuronal interactions and implicitly a code employed in the functional integration of brain systems Thenature of transients expressed conjointly in distinct neuronal populations repoundects the underlyingcoupling among populations This coupling may be synchronous (and possibly oscillatory) or asynchro-nous A critical distinction between synchronous and asynchronous coupling is that the former isessentially linear and the latter is nonlinear The nonlinear nature of asynchronous coupling enables therich context-sensitive interactions that characterize real brain dynamics suggesting that it plays a role infunctional integration that may be as important as synchronous interactions The distinction betweenlinear and nonlinear coupling has fundamental implications for the analysis and characterization ofneuronal interactions most of which are predicated on linear (synchronous) coupling (eg cross-correlograms and coherence) Using neuromagnetic data it is shown that nonlinear (asynchronous)coupling is in fact more abundant and can be more signicentcant than synchronous coupling

Keywords neuronal transients complexity functional integration neural codes selectionself-organization

1 INTRODUCTION

This paper is about the dynamical aspects of brainfunction Brain states are inherently labile with acomplexity and transience that renders their invariantcharacteristics elusive The position adopted in thesearticles is that the most fruitful approach to under-standing brain dynamics is to study this instability andtransience The aim of this paper is to introduce thenotion of neuronal transients and the underlying frame-work within which issues such as neuronal couplingneuronal codes functional integration self-organizationand the special complexity of brain dynamics can beaddressed The central tenet is that the behaviour ofneuronal systems can be viewed as a succession oftransient spatio-temporal patterns of activity thatmediate adaptive perceptual synthesis and sensorimotorintegration This integration is shaped by the brainrsquosanatomical infrastructure principally connections thathas been selected to ensure the adaptive nature of thedynamics that ensue Although rather obvious thisformulation embodies one fundamental point namelythat any proper description of brain dynamics should

have an explicit temporal dimension In other wordsmeasures of brain activity are only meaningful whenspecicented over extended periods of time Simply appre-ciating this fact leads to quite compelling insights aboutbrain organization and places some extant concepts in amore general context The centrst example considered inthis paper is that of neuronal codes when trying toconstruct a taxonomy of neuronal codes it becomes clearthat existing formulations are special cases of a moregeneric transient coding This is particularly important inrelation to fast dynamic interactions among neuronalpopulations that are characterized by synchronySynchronization has become popular in the past years(eg Eckhorn et al 1988 Gray amp Singer 1989 Engel et al1991) and yet may represent only one domain in thepossible and as will be shown actual universe of interac-tions Transient coding subsumes both synchronous andasynchronous interactions and it is the latter whichmediate the nonlinear and context-sensitive features ofbrain dynamics

The arguments presented in these papers depend inpart on a mathematical formulation that is developed toreinforce illustrate and at times motivate the ideas

Phil Trans R Soc Lond B (2000) 355 215^236 215 copy 2000 The Royal SocietyReceived 2 September 1998 Accepted 8 June 1999

introduced Important mathematical derivations areprovided in the appendices for the interested readerwhile only key equations are presented in the main textIn general it is the form of these equations that isimportant not their content Details concerning dataacquisition and simulation parameters are provided inthe centgure legends This paper is divided into six sectionsIn frac12 2 we review the conceptual and mathematical basisof neuronal transients This section uses a fundamentalequivalence between two mathematical formulations ofnonlinear systems to show that descriptions of braindynamics in terms of (i) neuronal transients and (ii) theeiexclective connectivity among interacting brain systems iscomplete and sucurrencient In frac12 3 the ensuing framework isused to motivate a taxonomy of putative neuronal codesthe relationships among them and the predictions thatarise In frac12 4 we review the evidence for neuronal transi-ents in terms of phenomena such as `dynamic correlationsrsquoand nonlinear interactions between brain regionsevidenced by asynchronous coupling This sectionconcludes with a direct test of the transient hypothesisusing data acquired with magnetoencephalography(MEG) that is based on the predictions from frac12 2 Section5 addresses the general relationship between asynchro-nous coupling and nonlinear interactions leading to adiscussion in frac12 6 of the neurobiological mechanisms (egmodulatory eiexclects) that might mediate them

2 NEURONAL TRANSIENTS

(a) Neuronal transients and timeThe assertion that teleologically meaningful measures

or metrics of brain dynamics have an explicit temporaldomain is neither new nor contentious (eg Von derMalsburg 1981 Optican amp Richmond 1987 Engel et al1991 Aertsen et al 1994 Freeman amp Barrie 1994 Abeleset al 1995 deCharms amp Merzenich 1996) A straight-forward analysis demonstrates its veracity the brain is ahighly nonlinear spatially extended system that is uniquein relation to other complex systems by virtue of itsconnectedness The brainrsquos architecture can be regardedas an ensemble of connections where the nature andorganization of these connections entails the substance ofthe system The signals that traverse connections (axonsand dendritic cell processes) do so in a centnite amount oftime Suppose that one wanted to posit a sucurrencient metricthat described the brain as a dynamical system in termsof neuronal activity A natural choice would be the statevariable x in a state equation

x(t)t ˆ f (x C) (1)

where x is a large vector of activities for each unit inthe brain These activities could be expressed in manyways for example centring at the initial segment of anaxon or local centeld potentials of neuronal populationsEquation (1) simply says that the change in activity withtime x(t) t is a function of x and C a collection ofcontrol parameters corresponding to the underlying time-invariant connection strengths (eg synaptic ecurrencacy)However equation (1) would not be sucurrencient because itmay take several possibly tens of milliseconds for theactivity in one neuron or population to propagate to its

recipient So the change in any unit is a function not just ofactivity elsewhere at time t but at time t and in the recentpastThis leads to the equation

x(t) ˆ f (x(t iexcl u) C) (2)

Equation (2) is in principle a sucurrencient description ofbrain dynamics and involves the variable x(t7u) whichrepresents activity at all times u preceding the moment inquestion x(t7u) is simply a neuronal transient (albeit avery global one) The degree of transience depends on howfar back in time it is necessary to go to fully capture thebrainrsquos dynamics In less abstract terms if we wanted todetermine the behaviour of a cell in the primary visualcortex (V1) then we would need to know the activity of allconnected cells in the immediate vicinity (say within thesame cortical column) over the last millisecond or so Wewould also need to know the activity in distant sites likethe lateral geniculate nucleus (LGN) and higher corticalareas that send aiexclerents some ten or more millisecondsago In short we need the recent history of all inputs

Transients can be expressed in terms of centring rates (egchaotic oscillations Freeman amp Barrie 1994) or individualspikes (eg syncentre chains Abeles et al 19941995) In whatfollows we will assume that the relevant measurementspertain to those behaviours of cells that can inpounduence othercells There is a fundamental reason for this which willbecome apparent below The analysis above is not just amathematical abstraction it has very real implications ata number of levels for example the emergence of fastoscillatory interactions among simulated neuronalpopulations depends on the time-delays implicit in axonaltransmission and the time constants of postsynapticresponses Another slightly more subtle aspect of thisformulation is that changes in synaptic ecurrencacy such asshort-term potentiation or depression take some time tobe mediated by intracellular mechanisms This means thatthe interaction between x(t7u) and C that models theseactivity-dependent eiexclects in equation (2) again dependson the relevant history of activity

(b) Diiexclerent levels of descriptionAn alternative perspective on the necessity of going

back in time to acquire a sucurrencient description ofneuronal dynamics is buried in the phrase above à sucurren-cient metric that describes the brain as a dynamicalsystem in terms of neuronal activityrsquo This perspective is alittle abstract but provides a strong basis for neuronaltransients By restricting ourselves to measuring neuronalactivity there are a vast number of critical variables thatare being ignored (eg the electro- and biochemical stateof every cell process in the brain) If we knew every one ofthem then equation (1) might be a tenable model consti-tuting a microscopic level of description that would beentirely sucurrencient and complete However because we donot have access to this complete ensemble of `hiddenrsquovariables we are apparently unable to ever describe braindynamics properly This is not necessarily the case

(i) A fundamental equivalenceAssume that every neuron in the brain is modelled by

some immensely complicated nonlinear dynamical systemof the sort described by equation (1) where the state

216 K J Friston The labile brain

PhilTrans R Soc Lond B (2000)

variables range from depolarization at every point in thedendritic tree to the phosphorylation status of everyrelevant enzyme The input to this system corresponds toaiexclerent activity and the output to centring at the cellrsquos initialsegment Notice that both the input and the output arehomologous in that they both measure that aspect of asystemrsquos (cellrsquos) behaviour that can inpounduence anothersystem Under these assumptions it can be shown that theoutput is a function of the recent history of its inputsFurthermore this relationship can be expressed as aVolterra series of the inputs (see Appendix A and frac12 2(c))The critical thing here is that we never need to know theunderlying and `hiddenrsquo variables that describe the detailsof each cellrsquos electrochemical and biochemical status weonly need to know the history of its inputs which ofcourse are the outputs of other cells (including the one inquestion)This leads to a conceptual model of the brain as acollection of dynamical systems (eg cells or populations ofcells) each of which is represented as an input^stateôutput model where the state remains for us foreverhidden However the inputs and outputs are accessible andare causally related where in this special case of massivelyconnected systems the output of one system constitutes theinput to another A complete description thereforecomprises the nature of these relationships (the Volterraseries) and the neuronal transients (past history of allinputs) This constitutes a mesoscopic level of descriptionwhich allows a certain degree of `black-boxnessrsquoas long asthere is no loss of information or precision in specifying theinteractions among the black boxes (cells or populations)

The equivalence in terms of specifying the behaviour of aneuronal system between microscopic and mesoscopic levelsof description is critical and one that is central to this paperand neuronal transients In short the equivalencemeans thatall the information inherent in the unobservable microscopicvariables that determine the response of a neuronal system isembodied in the history of its observable inputs and outputsThis means that neuronal transients are a sucurrencient descrip-tion of a system which eschew the measurement of hiddenvariables when predicting responses Although the micro-scopic level of description may be more causally inter-pretable from the point of view of response predictionneuronal transients are an equivalent representation

We have focused above on the distinction betweenmicroscopic and mesoscopic levels of description Themacroscopic level is reserved for approaches exemplicentedby synergistics (Haken 1983) that try to characterize thespatio-temporal evolution of brain dynamics in terms of asmall number of macroscopic order parameters (seeKelso (1995) for an engaging exposition) For examplemacroscopic variables can be extracted from large-scaleobservations such as magnetoencephalography (MEG)using the order parameter concept order parameters arecreated and determined by the cooperation of micro-scopic quantities and yet at the same time govern thebehaviour of the whole system We will not deal withthese approaches here but interested readers are referredto Jirsa et al (1995) for an example

(c) A nonlinear frameworkThe fact that a mesoscopic level of description exists

suggests that (i) a complete description of dynamics couldbe cast in terms of neuronal transients and (ii) a

complete model of eiexclective connectivity (ie the causalinpounduences that one neuronal system exerts over another)should take the form of a Volterra series These are quitefundamental conclusions The primary focus of thesepapers is the centrst conclusion namely one can either tryto measure every aspect of brain function and charac-terize the dynamics in terms of equation (1) or one canidentify the essential inputs and outputs of its componentsand work explicitly with their recent history ie framethe dynamics in terms of neuronal transients as in equa-tion (2) The former is impossible The latter is the subjectof this paper

Volterra series oiexcler a very general form for the func-tions in equation (2) and can be expressed as

xi(t) ˆ O0permilx(t)Š Dagger O1permilx(t)Š Dagger Dagger Onpermilx(t)Š Dagger (3)

where xi(t) is the activity of the ith unit OnpermilcentŠ is the nthorder Volterra operator and has associated with it akernel or smoothing function h that operates on the recenthistory of the inputs Volterra series are functional Taylorexpansions and are generally thought of as nonlinearconvolutions or polynomial expansions with memory (seeAppendix A) We have found this nonlinear system identi-centcation framework very useful when characterizingneuromagnetic and haemodynamic time-series fromfunctional magnetic resonance imaging (fMRI) and it isused below many times See also Stevens (1994)

The distinction and equivalence between microscopicand mesoscopic levels of description is illustrated incentgure 1 Clearly this equivalence cannot be demonstratedfor real neuronal systems but can be shown using reason-ably realistic synthetic or model systems Figure 1contains a schematic that summarizes the equationsbehind the simulations used in these papers (seeAppendix B) Collectively these equations are an exampleof equation (1) because they deal with all the relevantstate variables (depolarization channel concentgurationdischarge probability etc) and control parameters(synaptic ecurrencacies time constants etc) These equationsconstitute a microscopic level of description enabling oneto predict the evolution of the system given only its stateat one point in time This system of diiexclerential equationshas an entirely equivalent description in terms of theVolterra kernels that mediate between inputs and theresponses (the centrst few are shown in centgure 1b) Thesekernels can be applied to incoming transients to give theresponses without knowing the underlying state vari-ables All that is needed is the input over a period of time

For completeness it can be noted that the Volterraformulation based on the recent history of a systemrsquosinputs is conceptually related to temporal embeddingused to `reconstructrsquo the dynamics of a system given onlyone variable Temporal embedding involves using thecurrent value of the state variable and a succession ofvalues at a number of discrete times in the past SeeMuller-Gerking et al (1996) for a useful discussion of thisapproach in relation to the nonlinear characterization ofneuronal time-series

Arguments like those above suggest that the `neuronalmomentrsquo lasts for tens if not hundreds of milliseconds andthat the instantaneous behaviour of neuronal units cannotbe divorced from their immediate temporal context This

The labile brain K J Friston 217

Phil Trans R Soc Lond B (2000)

is not startling and is similar to noting that `populationcodesrsquo are necessarily distributed over many unitsNeuronal transients take this one step further and stipulatethat `transient codesrsquo are necessarily distributed over timeLooked at in this way neural transients are a naturalextension of the trend to characterize brain dynamics inrelation to the context in which they occur Neuronal tran-sients represent an attempt to generalize the notion ofpopulation dynamics into the temporal domain

(d) Eiexclective connectivity and Volterra kernelsThe second conclusion above (a complete model of

eiexclective connectivity should take the form of a Volterraseries) implies that a complete characterization of eiexclec-tive connectivity among neuronal systems can be framedin terms of the Volterra kernels associated with the trans-formations of and interactions among inputs that yieldthe outputs

(i) Eiexclective connectivityFunctional integration is usually inferred using correla-

tions among measurements of neuronal activity indiiexclerent brain systems In imaging neuroscience theterm `functional connectivityrsquo denotes the simple presenceof these correlations (Friston 1995a) However correla-tions can arise in a variety of ways For example inmulti-unit electrode recordings they can result fromstimulus-locked transients evoked by a common input orrepoundect stimulus-induced oscillations mediated by synapticconnections (Gerstein amp Perkel 1969 Gerstein et al 1989)Integration within a distributed system is better under-

stood in terms of èiexclective connectivityrsquo Eiexclectiveconnectivity refers explicitly to `the inpounduence that oneneural system exerts over another either at a synaptic(ie synaptic ecurrencacy) or population levelrsquo (Friston 1995a)It has been proposed (Aertsen amp PreiTHORNl 1991) that `the[electrophysiological] notion of eiexclective connectivityshould be understood as the experiment- and time-dependent simplest possible circuit diagram that wouldreplicate the observed timing relationships between therecorded neuronsrsquo This speaks to two important points(i) eiexclective connectivity is dynamic and (ii) it dependson a model of the interactions

If eiexclective connectivity is the inpounduence that one neuralsystem exerts over another it should be possible given theeiexclective connectivity and the aiexclerent activity to predictthe response of a recipient population This is preciselywhat Volterra kernels do Any model of eiexclective connec-tivity will be a special case of a Volterra series and anymeasure of eiexclective connectivity can be reduced to a set ofVolterra kernels An important aspect of eiexclective connec-tivity is its context sensitivity Eiexclective connectivity issimply the eiexclectrsquo that an input has on the output of atarget system This eiexclect will be sensitive to the history ofthe inputs (and outputs) and of course the microscopicstate and causal architecture intrinsic to the target popula-tionThis intrinsic dynamical structure is embodied in theVolterra kernels and the current state of the target popula-tion enters thought the history of the outputs that can re-enter as inputs In short Volterra kernels are synonymouswith eiexclective connectivity because they characterize themeasurable eiexclectrsquo that an input has on its target The use



(a)

inputs output outputx (t)

(b)

inputsx (t ndash 0)x (t ndash 1)x (t ndash 2)

Pjk

excitatory

inhibitory

discharge probability Dj

VrDj = s [ V ndash Vr]

transmembrane potential V

p[ channel opening]C para V para t = gk(V ndash Vk)S

open closed

channel kinetics

gkt

Pk = 1 ndash (1 ndash PjkDj)P

high-order convolution withVolterra kernels (hn)

h0 = 0

03

02

01

0

h1

time [ ms]0 40 80 120

120

80

40

h2

time [ ms]0 40 80 120

(1 ndash gk)Pk

Figure 1 Schematic illustrating the distinction between (a) a microscopic level of description of a neuronal system (eg cell orpopulation) where all the hidden variables are known enabling the output to be causally related to the instantaneous inputand (b) a more `black-boxrsquo mesoscopic level in which it is only necessary to know the recent history of the inputs to determine theoutputs The left panel details the operational equations of simulated neuronal populations used in subsequent sections andthe right panel gives an example of the Volterra kernels that characterize the ensuing inputôutput relationships The simulatedpopulations are described in detail in Appendix B and comprise two subpopulations (one excitatory and one inhibitory) Thesepopulations are described in terms of the mean transmembrane potential (V ) and the probability that constituent units ofsubpopulation j will centre (Dj) in a deterministic way The connectivity between them is described in terms of Pjk the probabilitythat a discharge event in j will open a postsynaptic channel in subpopulation k The probability of channel opening Pk iscomputed by considering all potential inputs including extrinsic inputs The probability of channel opening enters into acentrst-order kinetic model of channel concentguration for all k channel types The expected proportion of open channels gk in turnmediates changes in transmembrane potential though conductance changes and the depolarization relative to the equilibriumpotential Vk for that channel Finally the discharge probability is computed as a sigmoid function of V and the eiexclective reversalpotential Vr The same inputôutput behaviour can be emulated by convolving the recent history of the inputs with a series ofVolterra kernels of increasing order In the example shown zeroth- centrst- and second-order kernels are shown for the AMPAsimulations depicted in centgure 4 In this instance the input was taken to be the injected current and the output corresponded tothe simulated local centeld potential These kernels were estimated using least squares after expressing them in terms of basisfunctions (eighth-order discrete sine set over 128 ms) as described in Appendix E

of Volterra kernels in characterizing eiexclective connectivitywill be dealt with elsewhere

3 NEURONAL CODES

(a) Diiexclerent sorts of codeThe conjecture that functional integration may be

mediated by the mutual induction and maintenance ofstereotyped spatio-temporal patterns of activity (ieneuronal transients) in distinct neuronal populations waspresented in Friston (1995b 1997a) Functional integrationrefers here to the concerted interactions among neuronalpopulations that mediate perceptual binding sensori-motor integration and cognition It pertains to themechanisms of and constraints under which thedynamics of one population inpounduence those of another Ithas been suggested by many that integration amongneuronal populations uses transient dynamics thatrepresent a temporal codersquo A compelling proposal is thatpopulation responses encoding a percept become orga-nized in time through reciprocal interactions todischarge in synchrony (Milner 1974 Von der Malsburg1985 Singer 1994) The use of the term èncodingrsquo herespeaks directly to the notion of codes

A `codersquo is used here to mean a measurement or metricof neuronal activity that captures teleologically mean-ingful transactions among diiexclerent parts of the brain Noattempt is made to discern the meaning or content of aputative code All that is assumed is that a code or metricmust necessarily show some dependency when used toassess two interacting neuronal populations or brainareas In other words a neuronal code is a metric thatreveals interactions among neuronal systems by enablingsome prediction of the activity in one population giventhe activity in another Clearly from frac12 2 neuronaltransients represent the most generic form of code Giventhat neuronal transients have a number of attributes (egtheir duration mean level of centring predominantfrequency etc) any one of these attributes is a contenderfor a more parsimonious code Although the term code isnot being used to denote anything that `codesrsquo for some-thing in the environment it could be used to decentne someaspect of a sensory evoked transient that had a highmutual information with a sensory parameter that wasmanipulated experimentally (eg Optican amp Richmond1987 Tovee et al 1993)

Given the above decentnition the problem of identifyingpossible codes reduces to establishing which metrics aremutually predictive or statistically dependent whenapplied to two connected neuronal systems This is quitean important point and leads to a very clear formulationof what can and cannot constitute a code and thediiexclerent sorts of codes that might be considered Further-more using this operational decentnition the problem ofcentnding the right code(s) reduces to identifying the formof the Volterra operators in equation (3) for if we knowthese we can predict exactly what will ensue in any unitgiven the dynamics elsewhere Conversely it follows thatthe diiexclerent forms that equation (3) can take shouldspecify the various codes likely to be encountered We willreturn to this point in a subsequent section

To discuss the nature of neuronal transients in relationto codes a taxonomy is now introduced The most

general form of coding is considered to be transientcoding All other codes are special cases or special casesof special cases The most obvious special case of atransient is when that transient shrinks to an instant intime The associated codes will be referred to as instanta-neous codes that subsume temporal coding and ratecoding depending on the nature of the metric employedA more important distinction is whether two transients intwo neuronal systems are synchronous or asynchronousleading to the notion of synchronous codes and asynchro-nous codes This synchronization may in turn be oscilla-tory or not leading to oscillatory codes or non-oscillatorycodes Therefore oscillatory codes are a special case ofsynchronous codes that are themselves special cases oftransient codes This hierarchial decomposition is shownschematically in centgure 2 This taxonomy is now reviewedin more detail working from the special cases to themore general

(i) Instantaneous codes temporal and rate codingThe distinction between temporal coding and rate

coding (see Shadlen amp Newsome 1995 de Ruyter vanSteveninck et al 1997) centres on whether the precisetiming of individual spikes is sucurrencient to facilitatemeaningful neuronal interactions In temporal codingthe exact time at which an individual spike occurs is theimportant measure and the spike-train is considered as apoint process The term temporal coding is used here inthis restricted sense as opposed to designating codes thathave a temporal domain (eg Von der Malsburg 1985Singer 1994) There is a critical distinction betweeninstantaneous temporal codes and those that invoke sometemporal patterning of spikes over time The formerinclude for example the instantaneous phase relationshipbetween a spike and some reference oscillation Althoughin simple systems knowledge of this phase will allowsome prediction of responses in a target unit it would notbe sucurrencient for more nonlinear systems where one wouldneed to know the history of phase modulation Thesecond sort of temporal code is distributed over time andrepresents a transient code There are clear examples ofthese codes that have predictive validity for example theprimary cortical representation of sounds by the



neuronal codes

instantaneous(eg rate codes)

asynchronous(nonlinear)

synchronous(linear)

transient

oscillatory(phase-coding)

non-oscillatory

Figure 2 A simple taxonomy of neuronal codes wherecommonly appreciated forms of encoding are seen as specialcases of each other

coordination of action potential timing (deCharms ampMerzenich 1996) These codes depend on the relativetiming of action potentials and implicitly by appealing toan extended temporal frame of reference fall into theclass of transient codes A very good example of this isprovided by the work of de Ruyter van Steveninck et al(1997) who show that the temporal patterning of spike-trains elicited in poundy motion-sensitive neurons by naturalstimuli carry twice the amount of information as anequivalent (Poisson) rate code

Instantaneous rate coding considers spike-trains asstochastic processes whose centrst-order moments (ie meanactivity) provide a metric with which neuronal inter-actions are enacted These moments may be in terms ofspikes themselves or other compound events (eg theaverage rate of bursting Bair et al 1994) The essentialaspect of rate coding is that a complete metric would bethe average centring rates of all the systemrsquos components atone instant in time Interactions based on rate coding areusually assessed in terms of cross-correlations and manymodels of associative plasticity are predicated on thesecorrelated centring rates (eg Hebb 1949) In this paperinstantaneous rate codes are considered insucurrencient asproper descriptions of neuronal interactions because inthe absence of `hiddenrsquo microscopic variables they arenot useful This is because they predict nothing about acell or population response unless one knows the micro-scopic state of that cell or population

(ii) Synchronous codes oscillatory and non-oscillatory codesThe proposal most pertinent to these forms of coding

is that population responses participating in theencoding of a percept become organized in timethrough reciprocal interactions so that they come todischarge in synchrony (Von der Malsburg 1985 Singer1994) with regular periodic bursting Frequency-specicentcinteractions and synchronization are used synonymouslyin this paper It should be noted that synchronizationdoes not necessarily imply oscillations Howeversynchronized activity is usually inferred operationally byoscillations implied by the periodic modulation of cross-correlograms of separable spike-trains (eg Eckhorn etal 1988 Gray amp Singer 1989) or measures of coherencein multichannel electrical and neuromagnetic time-series(eg Llinas et al 1994) The underlying mechanism ofthese frequency-specicentc interactions is usually attributedto phase-locking among neuronal populations (egSporns et al 1992 Aertsen amp PreiTHORNl 1991) The key aspectof these metrics is that they refer to the extendedtemporal structure of synchronized centring patterns eitherin terms of spiking (eg syncentre chains Abeles et al 1994Lumer et al 1997) or oscillations in the ensuing popula-tion dynamics (eg Singer 1994)

One important aspect that distinguishes oscillatoryfrom non-oscillatory codes is that the former can embodyconsistent phase relationships that may play a mechanisticrole in the ontology of adaptive dynamics (eg Tononi etal 1992) This has been proposed for theta rhythms in thehippocampus and more recently for gamma rhythms (egBurgess et al 1994 see Jeiexclerys et al (1996) for furtherdiscussion)

Many aspects of functional integration and featurelinking in the brain are thought to be mediated by

synchronized dynamics among neuronal populations(Singer 1994) Synchronization repoundects the direct reci-procal exchange of signals between two populationswhereby the activity in one population inpounduences thesecond such that the dynamics become entrained andmutually reinforcing In this way the binding of diiexclerentfeatures of an object may be accomplished in thetemporal domain through the transient synchronizationof oscillatory responses (Von der Malsburg 1981) This`dynamical linkingrsquo decentnes their short-lived functionalassociation Physiological evidence is compatible with thistheory (eg Engel et al 1991) synchronization of oscilla-tory responses occurs within as well as among visualareas for example between homologous areas of the leftand right hemispheres and between areas at diiexclerentlevels of the visuomotor pathway (Engel et al 1991 Roelf-sema et al 1997) Synchronization in the visual cortexappears to depend on stimulus properties such as conti-nuity orientation and motion coherence Synchronizationmay therefore provide a mechanism for the binding ofdistributed features and contribute to the segmentation ofvisual scenes More generally synchronization mayprovide a powerful mechanism for establishing dynamiccell assemblies that are characterized by the phase andfrequency of their coherent oscillations

The problem with these suggestions is that there isnothing essentially dynamic about oscillatory interactionsAs argued by Erb amp Aertsen (1992) `the question mightnot be so much how the brain functions by virtue ofoscillations as most researchers working on cortical oscil-lations seem to assume but rather how it manages to doso in spite of themrsquo In order to establish dynamic cellassemblies it is necessary to create and destroy synchro-nized couplings It is precisely these dynamic aspects thatrender synchronization per se relatively uninteresting butspeak to changes in synchrony (eg Desmedt amp Tomberg1994) and the transitions between synchrony and asyn-chrony as the more pertinent phenomenon

(iii) Transient coding synchronous and asynchronous codesAn alternative perspective on neuronal codes is provided

by work on dynamic correlations (Aertsen et al 1994) asexemplicented inVaadia et al (1995) A fundamental phenom-enon observed by Vaadia et al (1995) is that followingbehaviourally salient events the degree of coherent centringbetween two neurons can change profoundly and systema-tically over the ensuing second or so One implication ofthis work is that a complete model of neuronal interactionshas to accommodate dynamic changes in correlationsmodulated on time-scales of 100^1000 ms A simple expla-nation for these dynamic correlations has been suggested(Friston 1995b) it was pointed out that the coexpression ofneuronal transients in diiexclerent parts of the brain couldaccount for dynamic correlations (see frac12 4) This transienthypothesis suggests that interactions are mediated by theexpression and induction of reproducible highly struc-tured spatio-temporal dynamics that endure over severalhundred milliseconds As in synchronization coding thedynamics have an explicit temporal dimension but there isno special dependence on oscillations or synchrony Inparticular the frequency structure of a transient in onepart of the brain may be very diiexclerent from that inanother In synchronous interactions the frequency



structures of both will be the same (whether they are oscil-latory or not)

If the transient model is correct then importanttransactions among cortical areas will be overlooked bytechniques that are predicated on rate coding(correlations covariance patterns spatial modes etc) orsynchronization models (eg coherence analysis andcross-correlograms) Clearly the critical issue here iswhether one can centnd evidence for asynchronous inter-actions that would render the transient level of thetaxonomy a useful one (see centgure 2) Such evidence wouldspeak to the importance of neuronal transients and placesynchronization in a proper context In frac12 4 we review theindirect evidence for neuronal transients and then providedirect evidence by addressing the relative contribution ofsynchronous and asynchronous coupling to interactionsbetween brain areas

4 THE EVIDENCE FOR NEURONAL TRANSIENTS

We are all familiar with neuronal transients in the formof evoked transients in electrophysiology However thecritical thing is whether transients in two neuronal popula-tions mediate their own induction For example at thelevel of multi-unit micro-electrode recordings correlationscan result from stimulus-locked transients evoked by acommon aiexclerent input or repoundect stimulus-induced inter-actionsoumlphasic coupling of neuronal assembliesmediated by synaptic connections (Gerstein amp Perkel1969 Gerstein et al 1989) The question here is whetherthese interactions can be asynchronous One importantindication that stimulus-induced interactions are notnecessarily synchronous comes from dynamic correlations

(a) Dynamic correlations and neuronal transientsAs mentioned above Vaadia et al (1995) presented

compelling results concerning neuronal interactions inmonkey cortex enabling them to make two fundamentalpoints (i) it is possible that cortical function is mediated bythe dynamic modulation of coherent centring amongneurons and (ii) that these time-dependent changes incorrelations can emerge without modulation of centring ratesOne implication is that a better metric of neuronal inter-actions could be framed in terms of dynamic changes incorrelations This possibility touches on the distinctionbetween temporal coding and rate coding as described inthe frac12 3 This distinction and the related debate (egShadlen amp Newsome 1995) centres on whether the precisetiming of individual spikes can represent sucurrencient infor-mation to facilitate information transfer in the brain Theposition adopted by Vaadia et al (1995) adds an extradimension to this debate while accepting that spike-trainscan be considered as stochastic processes (ie the exacttime of spiking is not vital) they suggest that temporalcoding may be important in terms of dynamic time-dependent and behaviourally specicentc changes in theprobability that two or more neurons will centre together InShadlen amp Newsome (1995) `precisersquo timing meanssynchronization within 1^5 ms In contrast Vaadia et al(1995) demonstrate looser coherence over a period of about100 ms (using 70 ms time bins) A simple explanation forthis temporally modulated coherence or dynamic correla-tion is provided by the notion of neuronal transients

Imagine that two neurons respond to an event with asimilar transient (a short-lived stereotyped time-dependent change in the propensity to centre) For example iftwo neurons respond to an event with decreased centring for400 ms and this decrease was correlated over epochs thenpositive correlations between the two centring rates would beseen for the centrst 400 of the epoch and then fade awaytherein emulating a dynamic modulation of coherence Inother words a transient modulation of covariance can beequivalently formulated as a covariance in the expressionof transients The generality of this equivalence can beestablished using singular value decomposition (SVD)Dynamic correlations are inferred on the basis of thecross-correlation matrix of the trial by trial activity as afunction of peristimulus time This matrix is referred to asthe joint peristimulus time histogram (JPSTH) andimplicitly discounts correlations due to stimulus-lockedtransients by dealing with correlations over trials (asopposed to time following the stimulus or event) Let xi bea matrix whose rows contain the activities recorded in uniti over a succession of time bins following the stimulus withone row for each trial Similarly for xj After these matriceshave been normalized the cross-correlation matrix isgiven by xT

i xj where Tdenotes transposition By noting theexistence of the singular value decomposition

xTI xj ˆ l1u

T1 v1 Dagger l2u

T2 v2 Dagger l3u

T3 v3 Dagger (4)

one observes that any cross-covariance structure xTi xj can

be expressed as the sum of covariances due to the expres-sion of paired transients (the singular vectors uk and vk)The expression of these transients covaries according tothe singular values lk In this model any observedneuronal transient in unit i is described by a linear combi-nation of the uk (or vi in unit j )

This is simply a mathematical device to show thatdynamic changes in coherence are equivalent to thecoherent expression of neural transients In itself it is notimportant in the sense that dynamic correlations are justas valid a characterization as neuronal transients andindeed may provide more intuitive insights into how thisphenomenon is mediated at a mechanistic level (egRiehle et al 1997) What is important is that the existenceof dynamic correlations implies the existence of transientsthat exist after stimulus-locked eiexclects have beendiscounted The next step is to centnd decentnitive evidencethat transients underpin asynchronous coupling orequivalently that coupled transients in two neuronalpopulations have a diiexclerent form or frequency structureThe essential issue that remains to be addressed iswhether a transient in one brain system that mediates theexpression of another transient elsewhere has the same ora diiexclerent temporal patterning of activity The impor-tance of this distinction will become clear below

(b) Synchrony asynchrony and spectral densitySynchronized fast dynamic interactions among

neuronal populations represent a possible mechanism forfunctional integration (eg perceptual binding) in thebrain but focusing on synchrony precludes a properconsideration of asynchronous interactions that may havean equally important and possibly distinct role In thissection the importance of synchronization is evaluated in



relation to the more general notion of neural transientsthat allow for both synchronous and asynchronous inter-actions Transients suggest that neuronal interactions aremediated by the mutual induction of stereotyped spatio-temporal patterns of activity among diiexclerent populationsIf the temporal structures of these transients are distinctand unique to each population then the prevalence ofcertain frequencies in one cortical area should predict theexpression of diiexclerent frequencies in another In contrastsynchronization models posit a coupled expression of thesame frequencies Correlations among diiexclerent frequen-cies therefore provide a basis for discriminating betweensynchronous and asynchronous coupling

Consider time-series from two neuronal populations orcortical areas The synchrony model suggests that theexpression of a particular frequency (eg 40 Hz) in onetime-series will be coupled with the expression of thesame (40 Hz) frequency in the other (irrespective of theexact phase relationship of the transients or whether theyare oscillatory or not) In other words the modulation ofthis frequency in one area can be explained or predictedby its modulation in the second Conversely asynchro-nous coupling suggests that the power at a referencefrequency say 40 Hz can be predicted by the spectraldensity in the second time-series at some frequenciesother than 40 Hz These predictions can be tested empiri-cally using standard time-frequency and regressionanalyses as exemplicented below These analyses are anextension of those presented in Friston (1997a) andconcentrm that both synchronous and asynchronouscoupling are seen in real neuronal interactions They useMEG data obtained from normal subjects whileperforming self-paced centnger movements These data werekindly provided by Klaus Martin Stephan and AndyIoannides

(c) Decentnitive evidence for asynchronous couplingAfter Laplacian transformation of multichannel

magnetoencephalographic data two time-series wereselected The centrst was an anterior time-series over thecentral prefrontal region and the second was a posteriorparietal time-series both slightly displaced to the leftThese locations were chosen because they had beenimplicated in a conventional analysis of responses evokedby centnger movements (Friston et al 1996) Figure 3 showsan example of these data in the time domain x(t) and inthe frequency domain g(t) following a time-frequencyanalysis See Appendix C for a description of time-frequency analyses and their relation to wavelet transfor-mations In brief they give the frequency structure of atime-series over a short period as a function of timeThe time-frequency analysis shows the dynamic changesin spectral density between 8 and 64 Hz over about 16 sThe cross-correlation matrix of the parietal andprefrontal time-frequency data is shown in centgure 3bThere is anecdotal evidence here for both synchronousand asynchronous coupling Synchronous coupling basedon the co-modulation of the same frequencies is manifestas hot spots along or near the leading diagonal of thecross-correlation matrix (eg around 20 Hz) More inter-esting are correlations between high frequencies in onetime-series and low frequencies in another In particularnote that the frequency modulation at about 34 Hz in the

parietal region (second time-series) could be explainedby several frequencies in the prefrontal region The mostprofound correlations are with lower frequencies in thecentrst time-series (26 Hz) but there are also correlationswith higher frequencies (54 Hz) and some correlationswith prefrontal frequencies around 34 Hz itself Theproblem here is that we cannot say whether there is atrue asynchronous coupling or whether there is simplesynchronous coupling at 34 Hz with other higher andlower frequencies being expressed in a correlated fashionwithin the prefrontal region These within-region correla-tions can arise from broad-band coherence (Bressler et al1993) or harmonics of periodic transients (see Jurgenset al 1995) In other words synchronous coupling at34 Hz might be quite sucurrencient to explain the apparentcorrelations between 34 Hz in the parietal region andother frequencies in the prefrontal region To address thisissue we have to move beyond cross-correlations andmake statistical inferences that allow for synchronouscoupling over a range of frequencies within the prefrontalarea This is eiexclected by treating the problem as a regres-sion analysis and asking whether the modulation of aparticular frequency in the parietal region can beexplained in terms of the modulation of frequencies in theprefrontal region starting with the model

g2(0t) ˆX

shy () pound g1(t) (5)

where g2(t) and g1(t) are the spectral densities fromthe parietal and prefrontal time-series and 0 is thefrequency in question (eg 34 Hz) shy () are the para-meters that have to be estimated To allow for couplingsamong frequencies that arise from the correlatedexpression of frequencies within the areas the predictoror explanatory variables g1(t) are decomposed intosynchronous and asynchronous predictors These corre-spond to the expression of the reference frequency in theprefrontal region g1(0t) and the expression of allthe remaining frequencies orthogonalized with respect tothe centrst predictor g1(t) (see Appendix D) By orthogo-nalizing the predictors in this way we can partition thetotal variance in parietal frequency modulation into thosecomponents that can be explained by synchronous andasynchronous coupling respectively

g2(0t) ˆ shy (0) pound g1(0t) DaggerX


Furthermore by treating one of the predictors as aconfound we can test statistically for the contribution ofthe other (ie either synchronous or asynchronous) usingstandard inferential techniques in this instance theF-ratio based on a multiple regression analysis of seriallycorrelated data (see Appendix D)

To recap for those less familiar with regression techni-ques we take a reference frequency in one time-series(eg the parietal region) and try to predict it using theexpression of all frequencies in the other (eg prefrontalregion) To ensure that we do not confuse asynchronousand synchronous interactions due to broad-band coher-ence and the like correlations with the referencefrequency within the prefrontal region are removed fromthe predictors



An example of the results of this sort of analysis areshown in centgure 3c Figure 3c(i) shows the proportion ofvariance that can be attributed to either synchronous(broken line) or asynchronous coupling (solid line) as afunction of frequency (0) In other words the proportionof variance in parietal spectral density that can bepredicted on the basis of changes in the same frequency inthe prefrontal region (broken line) or on the basis ofother frequencies (solid line) Figure 3c(ii) portrays thesignicentcance of these predictions in terms of the associatedp-values and shows that both synchronous and asynchro-nous coupling are signicentcant at 34 Hz (ie the middlepeak in centgure 3c(i)(ii))

In contrast the high correlations between 48 Hz in thesecond time-series and 26 Hz in the centrst is well away fromthe leading diagonal in the cross-correlation matrix withlittle evidence of correlations at either of these frequenciesalone The regression analysis concentrms that at thisfrequency asynchronous coupling prevails The only signicent-cant coupling is asynchronous (right peak in centgure 3c(i)(ii))and suggests that the expression of 48 Hz gamma activity inthe parietal region is mediated in part by asynchronousinteractions directly or vicariously with the prefrontalcortex Note that a common modulating source inpounduencingboth the parietal and prefrontal regions cannot be invokedas an explanation for this sort coupling because this eiexclectwouldbe expressed synchronously

(d) SummaryThe above example was provided to illustrate both

mixed and asynchronous coupling and to introduce theconcept that simply observing correlations betweendiiexclerent frequency modulations is not sucurrencient to inferasynchronous coupling Broad-band coherence in thecontext of oscillations leads naturally to cross-frequencycoupling and more importantly non-oscillatory butsynchronous interactions will as a matter of course intro-duce them by virtue of the tight correlations betweendiiexclerent frequencies within each area (eg Jurgens et al1995) By discounting these within time-series correlationsusing the orthogonalization above one can reveal anyunderlying asynchronous coupling Results of this sort arefairly typical (we have replicated them using diiexclerentsubjects and tasks) and provide decentnitive evidence forasynchronous coupling Generally our analyses show bothsynchronous and asynchronous eiexclects where the latter aretypically greater in terms of the proportion of varianceexplained As in the example presented here it is usual tocentnd both sorts of coupling expressed in the same data

In conclusion using an analysis of the statistical depen-dence between spectral densities measured at diiexclerentpoints in the brain the existence of asynchronouscoupling can be readily concentrmed It is pleasing that sucha simple analysis should lead to such an importantconclusion These results are consistent with transientcoding and imply that correlations (rate coding) andcoherence (synchrony coding) are neither complete norsucurrencient characterizations of neuronal interactions andsuggest that higher-order more general interactions maybe employed by the brain In the remaining sections theimportance of asynchronous interactions and theirmechanistic basis will be addressed using both simulatedand real neuronal time-series

5 COUPLING AND CODES

(a) Asynchronous coupling and nonlinear interactionsWhy is asynchronous coupling so important The

reason is that asynchronous interactions embody all thenonlinear interactions implicit in functional integrationand it is these that mediate the diversity and context-sensitive nature of neuronal interactions The nonlinearnature of interactions between cortical brain areasrenders the eiexclective connectivity among them inherentlydynamic and contextual Compelling examples ofcontext-sensitive interactions include the attentionalmodulation of evoked responses in functionally specia-lized sensory areas (eg Treue amp Maunsell 1996) andother contextually dependent dynamics (see Phillips ampSinger (1997) for an intriguing discussion)

One of the key motivations for distinguishing betweensynchronous and asynchronous coupling is that the under-lying mechanisms are fundamentally diiexclerent In brief itwill be suggested that synchronization emerges from thereciprocal exchange of signals between two populationswherein the activity in one population has a direct or`drivingrsquo eiexclect on the activity of the second In asynchro-nous coding the incoming activity from one populationmay exert a `modulatoryrsquo inpounduence not on the activity ofunits in the second but on the interactions among them(eg eiexclective connectivity) This indirect inpounduence will-lead to changes in the dynamics intrinsic to the secondpopulation that could mediate important contextual andnonlinear responses Before addressing the neural basis ofthese eiexclects in the next section the relationship betweenasynchrony and nonlinear coupling is established anddiscussed in relation to neuronal codes

Perhaps the easiest way to see that synchronized inter-actions are linear is to consider that the dynamics of anyparticular neuronal population can be modelled in termsof a Volterra series of the inputs from others If thisexpansion includes only the centrst-order terms then theFourier transform of the centrst-order Volterra kernelcompletely specicentes the relationship between the spectraldensity of input and output in a way that precludes inter-actions among frequencies or indeed inputs (as shownbelow) The very presence of signicentcant coupling betweenfrequencies above and beyond covariances between thesame frequencies implies a centrst-order approximation isinsucurrencient and by decentnition second- and higher-ordernonlinear terms are required In short asynchronouscoupling repoundects the nonlinear component of neuronalinteractions and as such is vital for a proper characteriza-tion of functional integration

To see more explicitly why asynchronous interactionsare so intimately related to nonlinear eiexclects considerequation (3) where for simplicity we focus on the eiexclectsof unit j on unit i discounting the remaining units

xi(t) ˆ O0permilxj(t)Š Dagger O1permilxj(t)Š Dagger O2permilxj(t)Š Dagger (7)

where xi(t) is the activity of one unit or population andxj(t) another By discounting the constant and high-orderterms we end up with a simple convolution model ofneuronal interactions

xi(t) ˆ O1permilxj(t)Š ˆ h1 Auml xj(t) (8)





2000

4000

6000

8000

10 000

12 000

14 000

16 000

time

(ms)

Hz

firs

t

ndash5 0

time-series

5 20 40

time-frequency

60frequency (Hz)

ndash5 0

time-series

5 20 40

time-frequency

60

(a)

(b)

(c)

10

prop

ortio

n of

var

ianc

e

02

015

01

005

0

20

30

40

50

60

20 40Hz second

60

(i)

p -va

lue

(unc

orre

cted

)

100

10ndash1

10ndash2

10ndash3

(ii)

20 40 60

frequency (Hz) (256 ms window)

Figure 3 Time-frequency analysis of MEG time-series from two remote cortical regions designed to characterize the relativecontribution of synchronous and asynchronous coupling in terms of correlated changes in spectral density within and amongfrequencies respectively Neuromagnetic data were acquired from normal subjects using a KENIKRON 37 channel MEG systemat 1 ms intervals for periods of up to 2 min During this time subjects were asked to make volitional joystick movements either inrandom directions or to the left every 2 s or so Epochs of data comprising 214 ms were extracted ECG artefacts were removedby linear regression and the data were transformed using a V3 transformation (ie Laplacian derivative (Ioannides et al 1990))to minimize spatial dependencies among the data Paired epochs were taken from a left prefrontal and left parietal region thatwere subsequently bandpass centltered (1^128 Hz) The data in this centgure come from a normal male performing leftwards move-ments (a) The two times-series x(t) (plots) and their corresponding time-frequency procentles g(t) (images) The centrsttime-series comes from the left prefrontal region roughly over the anterior cingulate and SMA The second comes from the leftsuperior parietal region The data have been normalized to zero mean and unit standard deviation The frequencies analysedwere 8 Hz to 64 Hz in 1 Hz steps (b) This is a simple characterization of the coupling among frequencies in the two regions andrepresents the cross-correlation matrix of the time-frequencies g(t) In this display format the correlation coecurrencients havebeen squared (c) These are the results of the linear regression analysis that partitions the amount of modulation in the second(parietal) time-series into components that can be attributed to synchronous (broken lines) and asynchronous (solid lines)contributions from the centrst (prefrontal) time-series (see main text and Appendix D) (i) The relative contribution in terms of theproportion of variance explained and (ii) in terms of the signicentcance using a semi-log plot of the corresponding p-values both as

where Auml denotes convolution and h1 is the centrst-orderVolterra kernel This can be expressed in the frequencydomain as

gi(t) ˆ jl()j2 pound gj(t) (9)

where l() is known as the transfer function and issimply the Fourier transform of the centrst-order kernel h1This equality says that the expression of any frequencyin unit i is predicted exactly by the expression of thesame frequency in unit j (after some scaling by thetransfer function) This is exactly how synchronous inter-actions have been characterized and furthermore is iden-tical to the statistical model employed to test forsynchronous interactions above In equation (6) theparameters shy (0) are essentially estimates of jl(0)j2 inequation (9) From this perspective the tests for asyn-chronous interactions in frac12 4 can be viewed as an implicittest of nonlinear interactions while discounting a linearmodel as a sucurrencient explanation for the observedcoupling See Erb amp Aertsen (1992) for an example oftransfer functions that obtain after the equationsdecentning a simulated neuronal population have beensimplicented to render them linear

In summary the critical distinction between synchro-nous and asynchronous coupling is the diiexclerence betweenlinear and nonlinear interactions among units or popula-tions Synchrony implies linearity The term generalizedsynchronyrsquo has been introduced to include nonlinear inter-dependencies (see Schiiexcl et al 1996) Generalizedsynchrony therefore subsumes synchronous and asynchro-nous coupling A very elegant method for makinginferences about generalized synchrony is described inSchiiexcl et al (1996) This approach is particularly interestingfrom our point of view because it implicitly uses the recenthistory of the dynamics through the use of temporalembedding to reconstruct the attractors analysedHowever unlike our approach based on a Volterra seriesformulation it does not explicitly partition the couplinginto synchronous and asynchronous components

(b) The taxonomy revisitedRelating synchrony and asynchrony directly to the

Volterra series formulation leads to a more formal andprincipled taxonomy of putative neuronal codes Recallthat the centrst level of the taxonomy distinguishes betweentransient codes and instantaneous codes In terms of theVolterra model the latter are a special case ofequation (7) when all the Volterra kernels shrink to apoint in time In this limiting case the activity in one unit

is simply a nonlinear function of the instantaneousactivity in the other unit (ie a polynomial expansion)

xi(t) ˆ h0 Dagger h1 xj(t) Dagger h2 xj(t)2 Dagger (10)

All other cases enter under the rubric of transient codesThese can be similarly decomposed into those that includenonlinear terms (asynchronous) and those that do not(synchronous) The centnal level of decomposition is ofsynchronous interactions into oscillatory and non-oscillatory The former is a special case where the transferfunction l() shrinks down on to one particularfrequency (For completeness it should be noted thatoscillatory codes expressed as kernels of any order thatexist predominantly at one frequency may exist GGreen personal communication)

In this framework it can be seen that the most impor-tant distinction that emerges after discounting special orlimiting cases is that between asynchronous and synchro-nous coupling and the implicit contribution of nonlinearinteractions The presence of coupling among diiexclerentfrequencies demonstrated in frac12 4 speaks to the prevalenceof strong nonlinearities in the functional integration ofneuronal populations The nature of these nonlinearitiesis the focus of the rest of this paper

6 THE NEURAL BASIS OF ASYNCHRONOUS

INTERACTIONS

In Friston (1997b) it was suggested that from a neuro-biological perspective the distinction between asynchro-nous and synchronous interactions could be viewed in thefollowing way Synchronization emerges from the reci-procal exchange of signals between two populationswhere each `drivesrsquo the other such that the dynamicsbecome entrained and mutually reinforcing In asynchro-nous coding the aiexclerents from one population exert a`modulatoryrsquo inpounduence not on the activity of the secondbut on the interactions among them (eg eiexclectiveconnectivity or synaptic ecurrencacy) leading to a change inthe dynamics intrinsic to the second population In thismodel there is no necessary synchrony between theintrinsic dynamics that ensue and the temporal pattern ofmodulatory input An example of this may be the facilita-tion of high-frequency gamma oscillations among nearbycolumns in visual cortex by transient modulatory inputfrom the pulvinar Here the expression of low-frequencytransients in the pulvinar will be correlated with theexpression of high-frequency transients in visual cortexTo test this hypothesis one would need to demonstratethat asynchronous coupling emerges when extrinsic



Figure 3 (Cont) functions of frequency in the parietal region The dotted line in the latter corresponds to p ˆ 005 (uncorrectedfor the frequencies analysed) This particular example was chosen because it illustrates all three sorts of coupling (synchronousasynchronous and mixed) From inspection of the cross-correlation matrix it is evident that power in the beta range (20 Hz) inthe second time-series is correlated with similar frequency modulation in the centrst albeit at a slightly lower frequency Theresulting correlations appear just oiexcl the leading diagonal (broken line) on the upper left The graphs on the right show that theproportion of variance explained by synchronous and asynchronous coupling is roughly the same and in terms of signicentcancesynchrony supervenes In contrast the high correlations between 48 Hz in the second time-series and 26 Hz in the centrst are wellaway from the leading diagonal with little evidence of correlations within either of these frequencies The regression analysisconcentrms that at this frequency asynchronous coupling prevails The variation at about 34 Hz in the parietal region could beexplained by several frequencies in the prefrontal region A formal analysis shows that both synchronous and asynchronouscoupling coexist at this frequency (ie the middle peak in the graphs on the right)

connections are changed from driving connections tomodulatory connections Clearly this cannot be done inthe real brain However we can use computationaltechniques to create a biologically realistic model of inter-acting populations and test this hypothesis directly

(a) Interactions between simulated populationsTwo populations were simulated using the model

described in Appendix B This model simulated entireneuronal populations in a deterministic or analoguefashion based loosely on known neurophysiologicalmechanisms In particular we modelled three sorts ofsynapse fast inhibitory (GABA) fast excitatory (AMPA)

and slower voltage-dependent synapses (NMDA)Connections intrinsic to each population used onlyGABA- and AMPA-like synapses Simulated glutami-nergic extrinsic connections between the two populationsused either driving AMPA-like synapses or modulatoryNMDA-like synapses In these and the remaining simula-tions transmission delays for extrinsic connections werecentxed at 8 ms By using realistic time constants the charac-teristic oscillatory dynamics of each population wereexpressed in the gamma range

The results of coupling two populations with unidirec-tional AMPA-like connections are shown in centgure 4a interms of the simulated local centeld potentials (LFP)



(a)0

ndash10

ndash20

ndash30

ndash40

ndash50

ndash60

ndash70

tran

smem

bran

e po

tent

ial (

mV

)

(b)0

ndash10

ndash20

ndash30

ndash40

ndash50

ndash60

ndash70

tran

smem

bran

e po

tent

ial (

mV

)

200 400 600 800 1000

time (ms)

1200 1400 1600 1800 2000

Figure 4 Simulated local centeld potentials (LFP) of two coupled populations using two diiexclerent sorts of postsynaptic responses(AMPA and NMDA-like) to extrinsic inputs to the second population from the centrst These data were simulated using the modeldescribed in centgure 1 and Appendix B The dotted line shows the depolarization eiexclected by sporadic injections of current into thecentrst population The key thing to note is that under AMPA-like or driving connections the second population is synchronouslyentrained by the centrst (a) whereas when the connections are modulatory or voltage dependent (NMDA) the eiexclects are muchmore subtle and resemble a frequency modulation (b) For the AMPA simulation self-excitatory AMPA connections of 015 and006 were used for the centrst and second populations respectively with an AMPA connection between them of 006 For theNMDA simulation the self-excitatory connection was increased to 014 in the second population and the AMPA connectionbetween the populations was changed to NMDA-like with a strength of 06

Occasional transients in the driving population weresimulated by injecting a depolarizing current of the samemagnitude at random intervals (dotted line) The tightsynchronized coupling that ensues is evident Thisexample highlights the point that near-linear couplingcan arise even in the context of loosely coupled highlynonlinear neuronal oscillators of the sort modelled hereIt should be noted that the connection strengths had to becarefully chosen to produce this synchronous entrainingDriving connections do not necessarily engender synchro-nized dynamics a point that we will return to laterContrast these entrained dynamics under driving connec-

tions with those that emerge when the connection ismodulatory or NMDA-like (centgure 4b) Here there is nosynchrony and as predicted fast transients of an oscilla-tory nature are facilitated by the aiexclerent input from thecentrst population This is a nice example of asynchronouscoupling that is underpinned by nonlinear modulatoryinteractions between neuronal populations The nature ofthe coupling can be characterized more directly using thetime-frequency analysis (identical in every detail) appliedto the neuromagnetic data of the previous section

Figure 5 shows the analysis of the AMPA simulationand demonstrates very clear broad-band coherence with



1000

2000

3000

4000

5000

6000

7000

8000

tim

e (m

s)H

z fi

rst

ndash5 0

time-series

5 20 40

time-frequency

60frequency (Hz)

ndash5 0

time-series

5 20 40

time-frequency

60

(a)

(b)

(c)

10

prop

orti

on o

f va

rian

ce

1

06

08

04

02

0

20

30

40

50

60

20 40Hz second

60

(i)

p -va

lue

(unc

orre

cted

)

100

10ndash10

10ndash20

(ii)

20 40 60


Figure 5 Time-frequency and coupling analyses for the LFPs of the simulations employing AMPA-like connections The formatand underlying analyses of this centgure are identical to centgure 3 The key thing to note is that the cross-correlations are almostsymmetrical suggesting synchrony and extensive broad-band coherence Indeed most p-values for synchronous (linear) couplingwere too small to compute

most of the cross-correlations among diiexclerent frequencieslying symmetrically about the leading diagonalSynchrony accounts for most of the coupling both interms of the variance in frequency modulation(centgure 5c(i)) and in terms of signicentcance (centgure 5c(ii))Note that at some frequencies the synchronous couplingwas so signicentcant that the p-values were too small tocompute These results can now be compared to theequivalent analysis of the NMDA simulation (centgure 6)

In contradistinction the cross-correlation matrix looksmuch more like that obtained with the MEG data in

centgure 3 Both in terms of the variance and inferenceasynchronous coupling supervenes at most frequenciesbut as in the real data mixed coupling is also evidentThese results can be taken as a heuristic conformation ofthe hypothesis that modulatory in this case voltage-dependent interactions are sucurrenciently nonlinear toaccount for the emergence of asynchronous dynamics

In summary asynchronous coupling is synonymouswith nonlinear coupling Nonlinear coupling can beframed in terms of the modulation of intrinsic inter-actions within a cortical area or neuronal population by



1000

2000

3000

4000

5000

6000

7000

8000

tim

e (m

s)H

z fi

rst

ndash5 0

time-series

5 20 40

time-frequency

60frequency (Hz)

ndash5 0

time-series

5 20 40

time-frequency

60

(a)

(b)

(c)

10

prop

orti

on o

f va

rian

ce

08

06

04

02

0

20

30

40

50

60

20 40Hz second

60

(i)

p -va

lue

(unc

orre

cted

)

100

10ndash5

10ndash10

10ndash15(ii)

20 40 60frequency (Hz) (256 ms window)

Figure 6 As for centgure 5 but now for simulations employing voltage-dependent NMDA-like connections In contradistinction tocentgure 5 the coupling here includes some profoundly asynchronous (nonlinear) components involving frequencies in the gammarange implicated in the analyses of real (MEG) data shown in centgure 3 In particular note the asymmetrical cross-correlationmatrix and the presence of asynchronous and mixed coupling implicit in the p-value plots on the lower right

extrinsic input oiexclered by aiexclerents from other parts of thebrain This mechanism predicts that the modulation offast (eg gamma) activity in one cortical area can bepredicted by a (nonlinear function of ) activity in anotherarea This form of coupling is very diiexclerent from coher-ence or other metrics of synchronous or linear couplingand concerns the relationship between the centrst-orderdynamics in one area and the second-order dynamics(spectral density) expressed in another In terms of theabove NMDA simulation transient depolarization in themodulating population causes a short-lived increasedinput to the second These aiexclerents impinge on voltage-sensitive NMDA-like synapses with time constants (in themodel) of 100 ms These synapses open and slowly closeagain remaining open long after an aiexclerent volleyBecause of their voltage-sensitive nature this input willhave no eiexclect on the dynamics intrinsic to the secondpopulation unless there is already a substantial degree ofdepolarization If there is then through self-excitationand inhibition the concomitant opening of fast excitatoryand inhibitory channels will generally increase membraneconductance decrease the eiexclective membrane timeconstants and lead to fast oscillatory transients This iswhat we observe in centgure 4b

(b) Nonlinear interactions and frequency modulationThe above considerations suggest that modulatory

aiexclerents can mediate a change in the qualitative natureof the intrinsic dynamics though a nonlinear voltage-dependent eiexclect that can be thought of in terms of afrequency modulation of the intrinsic dynamics by thisinput This motivates a plausible model of the relation-ship between the intrinsic dynamics of one populationcharacterized by its spectral density g2(0t) and theactivity in the centrst population x1(t) that is mediated bythe modulatory eiexclects of the latter These eiexclects can bemodeled with aVolterra series

g2(0t) ˆ O0permilx1(t)Š Dagger O1permilx1(t)Š Dagger (11)

It is interesting to note that this relationship is a moregeneral version of the statistical model used to test forcoupling in the previous section namely equation (5)This is because a time-frequency analysis itself is a simpleform of a Volterra series (see Appendix C) The motiva-tion behind this particular form of coupling between tworegions is predicated on the mechanistic insights providedby simulations of the sort presented above

Given the dynamics of the two populations from theNMDA simulations we can now estimate the form andsignicentcance of the Volterra kernels in equation (11) tocharacterize more precisely the nature of the nonlinearcoupling In this case the Volterra kernels were estimatedusing ordinary least squares This involves a dimensionreduction and taking second-order approximations of theVolterra series expansion (see Appendix E for details)Inferences about the signicentcance of these kernels aremade by treating the least-squares estimation as a generallinear model in the context of a multiple regression forserially correlated data (Worsley amp Friston 1995) as forthe time-frequency analyses The results of this analysisfor each frequency 0 are estimates of the Volterrakernels themselves (centgure 7b) and their signicentcance ie

the probability of obtaining the observed data if thekernels were equal to zero (centgure 7a) By applying theestimated kernels to the activity in the centrst populationone can visualize the expected and actual frequencymodulation at w0 (centgure 7c)

It is clear that this approach picks up the modulation ofspecicentc frequency components in the second populationand furthermore the time constants (ie duration ortemporal extent of the Volterra kernels) are consistentwith the time constants of channel opening in the simula-tion in this case the long time constants associated withvoltage-dependent mechanisms Knowing the form of theVolterra kernels allows one to characterize the frequencymodulation elicited by any specicented input Figure 8 showsthe actual frequency structure of the modulated popula-tion in the NMDA simulation and that predicted on thebasis of the activity in the centrst population Figure 8dshows the complicated form of frequency modulation thatone would expect with a simple Gaussian input over afew hundred milliseconds

Of course the critical test here is to apply this analysisto the real data of frac12 5 and see if similar eiexclects can bedemonstrated They can A good example is presented incentgure 9 showing how a slow nonlinear function(modelled by the Volterra kernels) of prefrontal activityclosely predicts the expression of fast (gamma) fre-quencies in the parietal region The implied modulatorymechanisms which may underpin this eiexclect are entirelyconsistent with the anatomy laminar specicentcity and func-tional role attributed to prefrontal eiexclerents (Rockland ampPandya 1979 Selemon amp Goldman-Rakic 1988)

7 CONCLUSION

This paper has introduced some basic considerationspertaining to neuronal interactions in the brain framed interms of neuronal transients and nonlinear coupling Thekey points of the arguments developed in this paper follow

(i) Starting with the premise that the brain can berepresented as an ensemble of connected input^stateôutput systems (eg cellular compartmentscells or populations of cells) there exists anequivalent inputôutput formulation in terms of aVolterra-series expansion of each systemrsquos inputs thatproduces its outputs (where the outputs to onesystem constitute the inputs to another)

(ii) The existence of this formulation suggests that thehistory of inputs or neuronal transients and theVolterra kernels are a complete and sucurrencientspecicentcation of brain dynamics This is the primarymotivation for framing dynamics in terms ofneuronal transients (and using aVolterra formulationfor models of eiexclective connectivity)

(iii) The Volterra formulation provides constraints on theform that neuronal interactions and implicit codesmust conform to There are two limiting cases(i) when the neuronal transient has a very shorthistory and (ii) when high-order terms disappearThe centrst case corresponds to instantaneous codes(eg rate codes) and the second to synchronous inter-actions (eg synchrony codes)

(iv) High-order terms in the Volterra model of eiexclectiveconnectivity speak explicitly to nonlinear interactions



and implicitly to asynchronous coupling Asynchro-nous coupling implies coupling among theexpression of diiexclerent frequencies

(v) Coupling among the expression of diiexclerent frequen-cies is easy to demonstrate using neuromagneticmeasurements of real brain dynamics This impliesthat nonlinear asynchronous coupling is a prevalentcomponent of functional integration

(vi) High-order terms in the Volterra model of eiexclectiveconnectivity correspond to modulatory interactionsthat can be construed as a nonlinear eiexclect of inputsthat interact with the dynamics intrinsic to the reci-pient system This implies that driving connectionsmay be linear and engender synchronous inter-actions whereas modulatory connections being

nonlinear may cause and be revealed by asynchro-nous coupling

The latter sections of this paper have shown thatasynchronous coupling can account for a signicentcant andsubstantial component of interactions between brainareas as measured by neuromagnetic signals Asynchro-nous coupling of this sort implies nonlinear coupling andboth speak to the diiexclerential form of neuronal transientsthat are expressed coincidentally in diiexclerent brain areasThis observation has been extended by testing thehypothesis that a parsimonious and neurobiologicallyplausible mechanism of nonlinear coupling employsvoltage-dependent synaptic interactions This led to theprediction that the dynamics in one region can predict the



Fmax = 165 df 979 p = 0000

(a)

(b)

(c)

100

10ndash5

10ndash10

10ndash15

p -va

lue

(unc

orre

cted

)

0 20 40

frequency (Hz)

60 80

01

0

ndash01

ndash02

ndash03

250

200

150

100

50

tim

e (m

s)

50 100 150 200 250time (ms)

1000 2000 3000 4000time (ms)

5000 6000 7000 8000

Figure 7 Nonlinear convolution (ie Volterra kernel) characterization of the frequency modulation eiexclected by extrinsicmodulatory inputs on the fast intrinsic dynamics The analysis described in Appendix E was applied to the simulated LFPsshown in centgure 4 using NMDA-like connections These are the same time-series analysed in centgure 6 In this instance we havetried to centnd the Volterra kernels that best model the frequency modulation of the dynamics in the second simulated populationgiven the time-series of the centrst (a) Plot of the signicentcance of the Kernels as a function of frequency in the modulated (second)population For the most signicentcant eiexclects (at 25 Hz) the estimated centrst- and second-order kernels are shown in (b) Applyingthese kernels to the time-series of the centrst population (dotted lines in (c)) one obtains a modulatory variable (solid line) that bestpredicts the observed frequency modulation (bottom line in (c))

changes in the frequency structure (a metric of intrinsicdynamics) in another Not only is this phenomenon easilyobserved in real data but in many instances it is extremelysignicentcant In particular it was shown that a nonlinearfunction of prefrontal dynamics could account for a signicent-cant component of the frequency modulation of parietaldynamics It should be noted that dynamic changes inspectral density may arise spontaneously from metastabledynamics even in the absence of extrinsic input (seeFriston paper 2 this issue) However this does not aiexclectthe conclusions above because it has been shown that atleast some of the (parietal) frequency modulation can beexplained by extrinsic (prefrontal) inputs

The importance of these observations relates both to themechanisms of functional integration in the brain and tothe way that we characterize neuronal interactions In

particular these results stress the importance of asynchro-nous interactions that are beyond the scope of synchronyAlthough this conclusion is interesting from a theoreticalstandpoint in terms of identifying the right metric that issensitive to the discourse between diiexclerent brain areas italso has great practical importance in the sense that manyways of characterizing neuronal time-series are based onsynchronization or linear models of neuronal interactions(eg cross-correlograms principal component analysissingular value decomposition coherence analyses etc) Anappreciation that nonlinear eiexclects can supervene in termsof their size and signicentcance over linear eiexclects such ascoherence (see centgure 3) may be important to ensure thatwe are measuring the right things when trying to charac-terize functional integration The theoretical implicationsare far-reaching because they appeal directly to the



(a)

(c)

(d )

10

20

30

40

50

60

freq

uenc

y (H

z)

4

3

1

2

0

10

30

20

50

40

60

freq

uenc

y (H

z)

2000 4000time (ms)

6000 8000

(b)

2000 4000time (ms)

6000 8000

100 200 300 400 500time (ms)

600 700 800 900 1000

Figure 8 Characterization of the observed and predicted frequency modulation using the estimated Volterra kernels from theanalysis summarized in centgure 7 (a) Actual and (b) predicted time-frequency procentles of the LFP of the second population Thepredicted procentle was obtained using the frequency-specicentc kernel estimates applied to the LFP of the centrst population Thesekernels can be applied to any input for example the `syntheticrsquo transient in (c) (d ) The ensuing frequency modulation shows abiphasic suppression around 40 Hz activity with transient increases (with diiexclerent latencies and time constants) at 25 50 and55 Hz

context-sensitive nature of neuronal interactions Modula-tory eiexclects are probably central to the mechanisms thatmediate attentional changes in receptive centeld propertiesand more generally the incorporation of context whenconstructing a unitrsquos responses to sensory inputs (seePhillips amp Singer 1997) One of the reasons that we chosethe prefrontal and parietal regions in the MEG analyseswas that both these regions are thought to participate indistributed attentional systems (eg Posner amp Petersen1990) and indeed our own work with fMRI in humansubjects suggests a modulatory role for prefrontal^parietalprojections (BIumlchel amp Friston 1997) In paper 2 (Fristonthis issue) we relate neuronal transients to dynamical

systems theory and complexity to arrive at a view offunctional organization in the brain that embraces bothlinear and nonlinear interactions

KJF is funded by the Wellcome Trust I would like to thankNicki Roiexcle for help preparing this manuscript and Semir ZekiRichard Frackowiak Erik Lumer Dave Chawla ChristianBIumlchel Chris Frith Cathy Price and the reviewers for theirscienticentc input Some of this work was centrst presented orally atthe Fifth Dynamical Neuroscience Satellite Symposium ofSociety for Neuroscience in New Orleans in 1997 I would like toacknowledge the useful feedback and comments from thatmeeting



Fmax = 45 df 9164 p = 0001

(b)

(a)

(c)

100

10ndash3

10ndash2

10ndash1

10ndash4

10ndash5

p -va

lue

(unc

orre

cted

)

0 20 40

frequency (Hz)

60 80

03

02

01

0

ndash01

250

200

150

100

50

tim

e (m

s)

50 100 150 200 250time (ms)

40002000 6000 8000 10 000time (ms)

12 000 14 000 16 000

Figure 9 Nonlinear convolution (ie Volterra kernel) characterization of the frequency modulation of parietal dynamics thatobtains by treating the prefrontal time-series as an extrinsic modulatory input The format and underlying analyses of thiscentgure are identical to the analysis of simulated LFPs presented in centgure 7 (a) There is evidence here for profound frequencymodulation at 48 Hz that appears to be loosely associated with movement but not completely so The top traces in (c) representthe observed prefrontal time-series before and after nonlinear convolution with the estimated kernels on the upper right Themiddle trace is the parietal time-series centltered at 48 Hz and the lower trace is the electromyogram repoundecting muscle activityassociated with movement (b) Inspection of the centrst- and second-order kernels suggests that fast gamma (48 Hz) activity in theparietal region is modulated by the transient expression of alpha (8 Hz) activity in the prefrontal region

APPENDIX A NONLINEAR SYSTEM IDENTIFICATION

AND VOLTERRA SERIES

Neuronal and neurophysiological dynamics are inherentlynonlinear and lend themselves to modelling by nonlineardynamical systems However due to the complexity ofbiological systems it is dicurrencult to centnd analytical equa-tions that describe them adequately An alternative is toadopt a very general model (Wray amp Green 1994) Aconventional method for representing a nonlineardynamic system (eg a neuron) is an input^stateôutputmodel (Manchanda amp Green 1998) These models can beclassicented as either that of a fully nonlinear system (wherethe inputs can enter nonlinearly) or a linear-analyticalsystem with the form

s(t)t ˆ f1(s) Dagger f2(s) pound xj

xi ˆ f3(s)

where s is a vector of states (eg the electrochemical stateof every cell compartment) f1 f2 and f3 are nonlinearfunctions xj is the input (eg aiexclerent activity from unit j)and xi the output (eg eiexclerent activity from unit i)Simple extensions to this description accommodatemultiple inputs Using functional expansions Fliess et al(1983) have shown how more general nonlinear diiexclerent-ial models can be reduced to a linear-analytical formThe Fliess fundamental formula describes the causal rela-tionship between the outputs and the recent history of theinputs This relationship can be expressed as a Volterraseries Volterra series allow the output to be computedpurely on the basis of the past history of the inputswithout reference to the state vector The Volterra series isan extension of the Taylor series representation to coverdynamic systems and has the general form

xi(t) ˆ O0permilx(t)Š Dagger O1permilx(t)Š Dagger Dagger Onpermilx(t)Š Dagger (A2)

where On permilcentŠ is the nth order Volterra operator

Onpermilx(t)Š ˆX

j1

X

jn

Z1

0

Z1

0

h nj1jn (u1 un)

pound xj1 (t iexcl u1) xjn (t iexcl u2)du1 dun

x(t) ˆ [x1(t) x2(t) ] is the neuronal activity in all theother connected units h n

j1jn is the nth order Volterrakernel for units j1 jn It can be shown that these seriescan represent any analytical time-invariant system Forfully nonlinear systems the above expansion about thecurrent inputs can be considered as an approximationthat is locally correct If the inputs enter in a sucurrencientlynonlinear way the Volterra kernels will themselves changewith input (cf activity-dependent synaptic connections)something that will be developed in paper 2 (Friston thisissue) in terms of instability and complexity The Volterraseries has been described as a `polynomial series withmemoryrsquo and is more generally thought of as a high orderor `nonlinear convolutionrsquo of the inputs to provide anoutput See Bendat (1990) for a fuller discussion

From the present perspective the Volterra kernels areessential in characterizing the eiexclective connectivity orinpounduences that one neuronal system exerts over anotherbecause they represent the causal characteristics of the

system in question Volterra series provide central links toconventional methods of describing inputôutputbehaviour such as the time-frequency analyses used inthis paper See Manchanda amp Green (1998) for a fullerdiscussion of Volterra series in the context of neuralnetworks

APPENDIX B THE NEURONAL SIMULATIONS

The simulations used a biologically plausible model of thedynamics of either one or several neuronal populationsThe model was of a deterministic or analoguersquo sort (cfErb amp Aertsen 1992) whose variables pertain to thecollective probabilistic behaviour of subpopulations ofneurons The variables in this model represent theexpected transmembrane potentials over units in eachsubpopulation and the probability of various eventsunderlying changes in that mean Each population wasmodelled in terms of an excitatory and inhibitory sub-population (see Jeiexclerys et al (1996) for an overview ofthese architectures) whose expected (ie mean) trans-membrane potentials V1 and V2 were governed by thefollowing diiexclerential equations

C pound V1t ˆ g1e pound (V1 iexclVe) Dagger g1v pound (V1 iexclVe)Dagger g1i pound (V1 iexcl Vi) Dagger gl pound (V1 iexcl Vl)

C pound V2t ˆ g2e pound (V2 iexclVe) Dagger g2i pound (V2 iexclVi)Dagger gl pound (V2 iexclVl)

where C is the membrane capacitance (taken to be 1 mF)and g1e g1v and g1i are the expected proportion of excita-tory (AMPA-like and NMDA-like) and inhibitory(GABA-like) channels open at any one time over all exci-tatory units in the population Similarly for g2e and g2i inthe inhibitory population gl is a leakage conductance VeVi and Vl are the equilibrium potentials for the variouschannels and resting conditions respectively Channelconcentgurations were modelled using a two-compartmentcentrst-order model in which any channel could be open orclosed For any given channel type k

gkt ˆ (1 iexcl gk) pound Pk iexcl gkfrac12k (B2)

where Pk is the probability of channel opening and frac12k isthe time constant for that channel type (to model classicalneuromodulatory eiexclects frac12AMPA was replaced by frac12AMPA(17M) where M was 08 unless otherwise specicented) Pkwas determined by the probability of channel opening inresponse to one or more presynaptic inputs This is simplyone minus the probability it would not open

Pk ˆ 1 iexcl brvbar(1 iexcl Pjk pound Dj(t iexcl ujk)) (B3)

where Pjk is the conditional probability that a dischargeevent in subpopulation j would cause the channel to openand Dj(t) is probability of such an event Pjk represents themean synaptic ecurrencacy for aiexclerents from subpopulation jand can be thought of as a connection strength ujk is theassociated transmission delay The centnal expression closesthe loop and relates the discharge probability to theexpected transmembrane potential in equation (B1)

Dj ˆ frac14sup2fVj iexcl Vrg (B4)



) (A1)

(B1)

where frac14sup2fcentg is a sigmoid function frac14sup2fxg ˆ 1(1 Dagger exp( iexcl xsup2)) Vr can be likened to the reversalpotential For NMDA-like channels the channel openingprobability was voltage dependent

Pjv ˆ Pjv pound frac148fVj iexcl Vvg (B5)

where Pjv is the conditional probability of opening given a

presynaptic input from subpopulation j when the post-synaptic membrane is fully depolarized

Each simulation comprised a 4096 iteration burn infollowed by 4096 iterations to see the dynamics thatensued Each iteration corresponds to 1ms V1 was takenas an index of the simulated local centeld potentialVoltage-dependent connections were only used betweenthe excitatory subpopulations of distinct populations Theintrinsic excitatoryînhibitory inhibitoryêxcitatory andinhibitory^ inhibitory connections Pjk where all centxed at06 04 and 02 respectively Intrinsic excitatoryêxcitatory connections were manipulated to control thedegree of spontaneous oscillation in the absence of otherinputs Extrinsic connections were excitatoryêxcitatoryconnections employing either AMPA or NMDAsynapses and were specicented depending on the architec-ture of the system being modelled All extrinsic trans-mission delays were 8 ms Remaining model parameterswere

Ve ˆ 0 mV gl ˆ 160 ms

Vi ˆ iexcl100 mV frac12e ˆ 6 ms

Vl ˆ iexcl60 mV frac12v ˆ 100 ms

Vr ˆ iexcl20 mV frac12i ˆ 10 ms

Vv ˆ iexcl10 mV

Note that sodium or potassium channels are not explicitlymodelled here The nonlinear dependency of dischargeprobability on membrane potential is implicit in equation(B4)

APPENDIX C TIME-FREQUENCY ANALYSIS

AND WAVELET TRANSFORMATIONS

The time-frequency analyses in this paper employed stan-dard windowed Fourier transform techniques with a256 ms Hanning window MEG frequencies analysedranged from 8 to 64 Hz Using a continuous time formu-lation for any given time-series x(t) the [time-dependent]spectral density g(t) can be estimated as

g(t) ˆ js(t)j2

where s(t) ˆ x(t) Auml f(t)pound exp (iexcl jt)g

ˆZ

w(u)poundexp (iexcl jt)pound x(t iexcl u)du

is 2ordm times the frequency in question and j ˆp

iexcl1Here Auml denotes convolution and w(t) is some suitablewindowing function of length l A Hanning function (abell-shaped function) w(t) ˆ (1 iexcl cos (2ordmt(l Dagger 1)))=2Šwith l ˆ 256 iterations or milliseconds was used Thiswindowed Fourier transform approach to time-frequencyanalysis is interesting in the present context because it isa special case equation (A2) ie g(t) obtains from

applying a second-order Volterra operator to x(t) wherethe corresponding kernel is

h(u1u2) ˆ w(u1)pound w(u2)pound exp(iexcl ju1)pound exp(iexcl ju2) (C2)

(a) Relationship to wavelet analysesThis simple time-frequency analysis is closely related to

wavelet analyses particularly those using Morlet wave-lets In wavelet analyses the energy at a particular scale isgiven by

g(t) ˆ jx(t) Auml w(t)j2 C3)

where w(t) represents a family of wavelets For examplethe complex Morlet wavelet is

w(t) ˆ A pound exp(iexcl u22frac142) pound exp(iexcl jt) (C4)

where the wavelet family is decentned by a constant micro (typi-cally about 6) such that pound frac14 ˆ micro Intuitively an analysisusing Morlet wavelets is the same as a time-frequencyanalysis in equation (C1) but where the windowingfunction becomes narrower with increasing frequency(ie w(t) ˆ A pound exp(iexcl u22frac142)) The relative advantagesof Morlet transforms in terms of time-frequency resolu-tion are discussed inTallon-Baudry et al (1997)

APPENDIX D DETECTING NONLINEAR

INTERACTIONS IN TERMS OF ASYNCHRONOUS

COUPLING

Nonlinear coupling between two dynamical systems isoften dicurrencult to detect There are two basic approacheswhich repoundect the search for nonlinearities in general (seeMuller-Gerking (1996) for an excellent discussion) Thecentrst involves reconstructing some suitable state space andcharacterizing the dynamics using the ensuing trajectories(eg Schiiexcl et al 1996) The second is to test for thepresence of nonlinearities using standard inferential tech-niques to look at mutual predictability after discountinglinear coupling The time-frequency analysis presented inthis paper is an example of the latter and is based on theobservation that signicentcant correlations among diiexclerentfrequencies (after removing those that can be explainedby the same frequency) can only be mediated bynonlinear coupling

The time-dependent spectral density at of the time-series x1(t) and x2(t) are estimated as above withg1(t) ˆ js1(t)j2 Similarly for x2(t) (see Appendix C) Totest for synchronous (linear) and asynchronous(nonlinear) coupling at a reference frequency 0 g1(0t)is designated as the response variable with explanatoryvariables g2(0t) and g2(t) in a standard multilinearregression extended to account for serially correlatedvariables (Worsley amp Friston 1995) g2(t) representsmodulation at all frequencies after the modulation at 0has been covaried out

g2(t)ˆ g2(t)iexclg2(0t)poundpinvfg2(0t)gpoundg2(t) (D1)

pinvfcentg is the pseudo-inverse This renders the synchro-nous g2(0t) and asynchronous g2(t) explanatory vari-ables orthogonal To test for asynchronous coupling theexplanatory variables comprise g2(t)while g2(0t) istreated as a confound To test for synchronous coupling



(C1)

g2(0t) is treated as the regressor of interest and g2(t)

the confounds The ensuing multiple regressions give theappropriate sums of squares (synchronous and asynchro-nous) F-values and associated p-values as a function of0 In practice the spectral densities are subject to aquadratic root transform as a preprocessing step toensure the residuals are approximately multivariate Gaus-sian (Friston 1997a)

The inferences above have to be corrected for serialcorrelations in the residuals necessarily introducedduring the convolution implicit in the time-frequencyanalysis (Worsley amp Friston 1995) Under the null hypoth-esis these can be modelled by convolution with w(t)2 thesquare of the window employed This would be exactlyright if we used the spectral densities themselves and isapproximately right using the quadratic root transformThis approximation leads to a test that is no longer exactbut is still valid (ie slightly conservative)

APPENDIX E ESTIMATING VOLTERRA KERNELS

The problem of estimating Volterra kernels is not atrivial one We have adopted a standard least-squaresapproach This has the advantage of providing for statis-tical inference using the general linear model To do thisone must centrst linearize the problem Consider the second-order approximation of equation (A2) with centnite`memoryrsquo T where the input is a single time-series x(t)and the output is denoted by y(t)

y(t) ordm O0permilx(t)Š Dagger O1permilx(t)Š Dagger O2permilx(t)Š (E1)

The second step in making the estimates of h0 h1 and h2

more tractable is to expand the kernels in terms of a smallnumber P of temporal basis functions bi(u) This allows usto estimate the coecurrencients of this expansion usingstandard least squares

let h0 ˆ g0

h1(u1) ˆXp

iˆ 1

g1i bi(u1)

h2(u1u2) ˆXp

iˆ 1

Xp

jˆ 1

g2ijbi(u1) pound bj(u2)

Now decentne a new set of explanatory variables zi(t) thatrepresent the original time-series x(t) convolved with theith basis function Substituting these expressions into equa-tion (E1) and including an explicit error term e gives

y(t) ˆ g0 DaggerXp

iˆ 1

g1i zi(t) Dagger

Xp

iˆ1

Xp

jˆ 1

g2ijzi(t) pound zj(t) Dagger e (E2)

This is simply a general linear model with response vari-able y(t) the observed time-series and explanatory vari-ables 1 zi(t) and zi(t)pound zj(t) These explanatory variables(convolved time-series of the original explanatory vari-ables) constitute the columns of the design matrix Theunknown parameters are g0 g1 and g2 from which thekernel coecurrencients h0 h1 and h2 are derived using equation(E2) Having reformulated the problem in this way wecan now use standard analysis procedures developed forserially correlated time-series that employ the general

linear model (Worsley amp Friston 1995) These proceduresprovide parameter estimates (ie estimates of the basisfunction coecurrencients and implicitly the kernels themselves)and statistical inferences about the signicentcance of thekernels or more precisely the eiexclect that they mediateThe issues of serial correlations in the residuals are iden-tical to those described in Appendix D

REFERENCES

Abeles M Prut Y Bergman H amp Vaadia E 1994Synchronisation in neuronal transmission and its importancefor information processing In Temporal coding in the brain (edG Buzsaki R Llinas W Singer A Berthoz amp T Christen)pp 39^50 Berlin Springer

Abeles M Bergman H Gat I Meilijson I Seidmann ETishby N amp Vaadia E 1995 Cortical activity poundipsamong quasi-stationary states Proc Natl Acad Sci USA 928616^8620

Aertsen A amp PreiTHORNl H 1991 Dynamics of activity and connec-tivity in physiological neuronal networks In Non linear dynamicsand neuronal networks (ed H G Schuster) pp 281^302 NewYorkVCH Publishers

Aertsen A Erb M amp Palm G 1994 Dynamics of functionalcoupling in the cerebral cortex an attempt at a model-basedinterpretation Physica D 75 103^128

Bair W Koch C Newsome W amp Britten K 1994 Relatingtemporal properties of spike trains from area MT neurons tothe behaviour of the monkey InTemporal coding in the brain (edG Buzsaki R Llinas W Singer A Berthoz amp T Christen)pp 221^250 Berlin Springer

Bendat J S 1990 Nonlinear system analysis and identicentcation fromrandom data NewYork Wiley

Bressler S L Coppola R amp Nakamura R 1993 Episodicmulti-regional cortical coherence at multiple frequenciesduring visual task performance Nature 366 153^156

BIumlchel C amp Friston K J 1997 Modulation of connectivity invisual pathways by attention cortical interactions evaluatedwith structural equation modelling and fMRI Cerebr Cortex 7768^778

Burgess N Recce M amp OrsquoKeefe J 1994 A model of hippo-campal function Neural Network 7 1065^1081

deCharms R C amp Merzenich M M 1996 Primary corticalrepresentation of sounds by the coordination of action poten-tial timing Nature 381 610^613

de Ruyter van Steveninck R R Lewen G D Strong S PKoberie R amp BialekW 1997 Reproducibility and variabilityin neural spike trains Science 275 1085^1088

Desmedt J E amp Tomberg C 1994 Transient phase-locking of40 Hz electrical oscillations in prefrontal and parietal humancortex repoundects the process of conscious somatic perceptionNeurosci Lett 168 126^129

Eckhorn R Bauer R JordanW Brosch M KruseW MunkM amp Reitboeck H J 1988 Coherent oscillations a mechanismof feature linking in the visual cortex Multiple electrode andcorrelationanalysis in the cat Biol Cybern 60121^130

Engel A K Konig P amp Singer W 1991 Direct physiologicalevidence for scene segmentationby temporal coding Proc NatlAcad Sci USA 88 9136^9140

Erb M amp Aertsen A 1992 Dynamics of activity in biology-oriented neural network models stability analysis at lowcentring rates In Information processing in the cortex Experiments andtheory (ed A Aertsen amp V Braitenberg) pp 201^223 BerlinSpringer

Fliess M Lamnabhi M amp Lamnabhi-Lagarrigue F 1983 Analgebraic approach to nonlinear functional expansions IEEETrans Circuits Syst 30 554^570



(E2)

Freeman W amp Barrie J 1994 Chaotic oscillations and thegenesis of meaning in cerebral cortex In Temporal coding in thebrain (ed G Buzsaki R Llinas W Singer A Berthoz amp TChristen) pp 13^18 Berlin Springer

Fries P Roelfsema P R Engel A Konig P amp Singer W1997 Synchronization of oscillatory responses in visual cortexcorrelates with perception in inter-ocular rivalry Proc NatlAcad Sci USA 94 12 699^12704

Friston K J 1995a Functional and eiexclective connectivity inneuroimaging a synthesis Hum Brain Mapp 2 56^78

Friston K J 1995b Neuronal transients Proc R Soc Lond B 261401^405

Friston K J 1997a Another neural code NeuroImage 5 213^220Friston K J 1997b Transients metastability and neuronal

dynamics NeuroImage 5 164^171Friston K J Stephan K M Heather J D Frith C D

Ioannides A A Liu L C Rugg M D Vieth J KeberH Hunter K amp Frackowiak R S J 1996 A multivariateanalysis of evoked responses in EEG and MEG dataNeuroImage 3 167^174

Gerstein G L amp Perkel D H 1969 Simultaneously recordedtrains of action potentials analysis and functional interpreta-tion Science 164 828^830

Gerstein G L Bedenbaugh P amp Aertsen A M H J 1989Neuronal assemblies IEEE Trans Boimed Engng 36 4^14

Gray C M amp Singer W 1989 Stimulus specicentc neuronal oscil-lations in orientation columns of cat visual cortex Proc NatlAcad Sci USA 86 1698^1702

Haken H 1983 Synergistics an introduction 3rd edn BerlinSpringer

Hebb D O 1949 The organization of behaviour NewYork WileyIoannides A A Hasson R amp Miseldine G J 1990 Model-

dependent noise elimination and distributed source solutionsfor the biomagnetic inverse problem SPIE 1351 471

Jeiexclerys J G R Traub R D amp Whittington M A 1996Neuronal networks for induced `40 Hzrsquo rhythms TrendsNeurosci 19 202^208

Jirsa V K Friedrich R amp Haken H 1995 Reconstruction ofthe spatio-temporal dynamics of a human magnetoencephalo-gram Physica D 89 100^122

Jurgens E Rosler F Hennighausen E amp Heil M 1995Stimulus induced gamma oscillations harmonics of alphaactivity NeuroReport 6 813^816

Kelso J A S 1995 Dynamic patterns the self-organisation of brainand behaviour Cambridge MA MIT Press

Llinas R Ribary U Joliot M amp Wang X-J 1994 Contentand context in temporal thalamocortical binding In Temporalcoding in the brain (ed G Buzsaki R Llinas W Singer ABerthoz amp T Christen) pp 251^272 Berlin Springer

Lumer E D Edelman G M amp Tononi G 1997 Neuraldynamics in a model of the thalamocortical system II Therole of neural synchrony tested through perturbations of spiketiming Cerebr Cortex 7 228^236

Manchanda S amp Green G G R 1999 The inputôutputbehaviour of dynamic neural networks Physica D(Submitted)

Milner P M 1974 A model for visual shape recognition PsycholRev 81 521^535

Muller-Gerking J Martinerie J Neuenschwander S PezardL Rebault B amp Varela F J 1996 Detecting non-linearitiesin neuro-electric signals a study of synchronous local centeldpotentials Physica D 94 65^81

Optican L amp Richmond B J 1987 Temporal coding of two-dimensional patterns by single units in primate inferior cortexII Informationtheoretic analysisJ Neurophysiol 57132^146

Phillips W A amp Singer W 1997 In search of common founda-tions for cortical computation Behav Brain Sci 4 657^683

Posner M I amp Petersen S E 1990 The attention system of thehuman brain A Rev Neurosci 13 25^42

Riehle A Grun S Diesmann M amp Aertsen A 1997 Spikesynchronization and rate modulation diiexclerentially involved inmotor cortical function Science 278 1950^1953

Rockland K S amp Pandya D N 1979 Laminar origins andterminations of cortical connections of the occipital lobe inthe rhesus monkey Brain Res 179 3^20

Roelfsema P R Konig P Engel A K Sireteanu R amp SingerW 1994 Reduced synchronization in the visual cortex of catswith strabismic amblyopia Eur J Neurosci 6 1645^1655

Schiiexcl S J So P Chang T Burke R E amp Sauer T 1996Detecting dynamical interdependence and generalizedsynchrony though mutual prediction in a neuronal ensemblePhys Rev E 54 6708^6724

Selemon L D amp Goldman-Rakic P S 1988 Common corticaland subcortical targets of the dorsolateral prefrontal andposterior parietal cortices in the rhesus monkey evidence fora distributed neural network subserving spatially guidedbehavior J Neurosci 8 4049^4068

Shadlen M N amp NewsomeW T 1995 Noise neural codes andcortical organization Curr Opin Neurobiol 4 569^579

Singer W 1994 Time as coding space in neocortical processinga hypothesis In Temporal coding in the brain (ed G Buzsaki RLlinas W Singer A Berthoz amp T Christen) pp 51^80Berlin Springer

Sporns O Gally J A Reeke G N amp Edelman G M 1989Reentrant signalling among simulated neuronal groups leadsto coherence in their oscillatory activity Proc Natl Acad SciUSA 86 7265^7269

Stevens C F 1994 What form should a cortical theory take InLarge-scale neuronal theories of the brain (ed C Kock amp J LDavies) pp 239^255 Cambridge MA MIT Press

Tononi G Sporns O amp Edelman G M 1992 Re-entry andthe problem of integrating multiple cortical areas simulationof dynamic integration in the visual system Cerebr Cortex 2310^335

Tovee M J Rolls ETTreves A amp Bellis R P1993Informationencoding and the response of single neurons in the primatetemporalvisual cortexJ Neurophysiol70 640^654

Treue S amp Maunsell H R 1996 Attentional modulation ofvisual motion processing in cortical areas MT and MSTNature 382 539^541

Vaadia E Haalman I Abeles M Bergman H Prut YSlovin H amp Aertsen A 1995 Dynamics of neuronal inter-actions in monkey cortex in relation to behavioural eventsNature 373 515^518

Von der Malsburg C 1981 The correlation theory of the brainInternal report Max-Planck-Institute for BiophysicalChemistry Gottingen Germany

Von der Malsburg C 1985 Nervous structures with dynamicallinks Ber Bunsenges Phys Chem 89 703^710

Worsley K J amp Friston K J 1995 Analysis of fMRI time-series revisitedoumlagain NeuroImage 2 173^181

Wray J amp Green G G R 1994 Calculation of the Volterrakernels of non-linear dynamic systems using an articentcialneuronal network Biol Cybern 71 187^195



introduced Important mathematical derivations areprovided in the appendices for the interested readerwhile only key equations are presented in the main textIn general it is the form of these equations that isimportant not their content Details concerning dataacquisition and simulation parameters are provided inthe centgure legends This paper is divided into six sectionsIn frac12 2 we review the conceptual and mathematical basisof neuronal transients This section uses a fundamentalequivalence between two mathematical formulations ofnonlinear systems to show that descriptions of braindynamics in terms of (i) neuronal transients and (ii) theeiexclective connectivity among interacting brain systems iscomplete and sucurrencient In frac12 3 the ensuing framework isused to motivate a taxonomy of putative neuronal codesthe relationships among them and the predictions thatarise In frac12 4 we review the evidence for neuronal transi-ents in terms of phenomena such as `dynamic correlationsrsquoand nonlinear interactions between brain regionsevidenced by asynchronous coupling This sectionconcludes with a direct test of the transient hypothesisusing data acquired with magnetoencephalography(MEG) that is based on the predictions from frac12 2 Section5 addresses the general relationship between asynchro-nous coupling and nonlinear interactions leading to adiscussion in frac12 6 of the neurobiological mechanisms (egmodulatory eiexclects) that might mediate them

2 NEURONAL TRANSIENTS

(a) Neuronal transients and timeThe assertion that teleologically meaningful measures

or metrics of brain dynamics have an explicit temporaldomain is neither new nor contentious (eg Von derMalsburg 1981 Optican amp Richmond 1987 Engel et al1991 Aertsen et al 1994 Freeman amp Barrie 1994 Abeleset al 1995 deCharms amp Merzenich 1996) A straight-forward analysis demonstrates its veracity the brain is ahighly nonlinear spatially extended system that is uniquein relation to other complex systems by virtue of itsconnectedness The brainrsquos architecture can be regardedas an ensemble of connections where the nature andorganization of these connections entails the substance ofthe system The signals that traverse connections (axonsand dendritic cell processes) do so in a centnite amount oftime Suppose that one wanted to posit a sucurrencient metricthat described the brain as a dynamical system in termsof neuronal activity A natural choice would be the statevariable x in a state equation

x(t)t ˆ f (x C) (1)

where x is a large vector of activities for each unit inthe brain These activities could be expressed in manyways for example centring at the initial segment of anaxon or local centeld potentials of neuronal populationsEquation (1) simply says that the change in activity withtime x(t) t is a function of x and C a collection ofcontrol parameters corresponding to the underlying time-invariant connection strengths (eg synaptic ecurrencacy)However equation (1) would not be sucurrencient because itmay take several possibly tens of milliseconds for theactivity in one neuron or population to propagate to its

recipient So the change in any unit is a function not just ofactivity elsewhere at time t but at time t and in the recentpastThis leads to the equation

x(t) ˆ f (x(t iexcl u) C) (2)

Equation (2) is in principle a sucurrencient description ofbrain dynamics and involves the variable x(t7u) whichrepresents activity at all times u preceding the moment inquestion x(t7u) is simply a neuronal transient (albeit avery global one) The degree of transience depends on howfar back in time it is necessary to go to fully capture thebrainrsquos dynamics In less abstract terms if we wanted todetermine the behaviour of a cell in the primary visualcortex (V1) then we would need to know the activity of allconnected cells in the immediate vicinity (say within thesame cortical column) over the last millisecond or so Wewould also need to know the activity in distant sites likethe lateral geniculate nucleus (LGN) and higher corticalareas that send aiexclerents some ten or more millisecondsago In short we need the recent history of all inputs

Transients can be expressed in terms of centring rates (egchaotic oscillations Freeman amp Barrie 1994) or individualspikes (eg syncentre chains Abeles et al 19941995) In whatfollows we will assume that the relevant measurementspertain to those behaviours of cells that can inpounduence othercells There is a fundamental reason for this which willbecome apparent below The analysis above is not just amathematical abstraction it has very real implications ata number of levels for example the emergence of fastoscillatory interactions among simulated neuronalpopulations depends on the time-delays implicit in axonaltransmission and the time constants of postsynapticresponses Another slightly more subtle aspect of thisformulation is that changes in synaptic ecurrencacy such asshort-term potentiation or depression take some time tobe mediated by intracellular mechanisms This means thatthe interaction between x(t7u) and C that models theseactivity-dependent eiexclects in equation (2) again dependson the relevant history of activity

(b) Diiexclerent levels of descriptionAn alternative perspective on the necessity of going

back in time to acquire a sucurrencient description ofneuronal dynamics is buried in the phrase above `a sucurren-cient metric that describes the brain as a dynamicalsystem in terms of neuronal activityrsquo This perspective is alittle abstract but provides a strong basis for neuronaltransients By restricting ourselves to measuring neuronalactivity there are a vast number of critical variables thatare being ignored (eg the electro- and biochemical stateof every cell process in the brain) If we knew every one ofthem then equation (1) might be a tenable model consti-tuting a microscopic level of description that would beentirely sucurrencient and complete However because we donot have access to this complete ensemble of `hiddenrsquovariables we are apparently unable to ever describe braindynamics properly This is not necessarily the case

(i) A fundamental equivalenceAssume that every neuron in the brain is modelled by

some immensely complicated nonlinear dynamical systemof the sort described by equation (1) where the state


























(a)


(b)


Pjk

excitatory

inhibitory





open closed

channel kinetics

gkt



h0 = 0

03

02

01

0

h1

time [ ms]0 40 80 120

120

80

40

h2

time [ ms]0 40 80 120

(1 ndash gk)Pk



3 NEURONAL CODES











neuronal codes



synchronous(linear)

transient


non-oscillatory






















xTI xj ˆ l1u

T1 v1 Dagger l2u

T2 v2 Dagger l3u

T3 v3 Dagger (4)













g2(0t) ˆX



























2000

4000

6000

8000

10 000

12 000

14 000

16 000

time

(ms)

Hz

firs

t

ndash5 0

time-series

5 20 40

time-frequency

60frequency (Hz)

ndash5 0

time-series

5 20 40

time-frequency

60

(a)

(b)

(c)

10

prop

ortio

n of

var

ianc

e

02

015

01

005

0

20

30

40

50

60

20 40Hz second

60

(i)

p -va

lue

(unc

orre

cted

)

100

10ndash1

10ndash2

10ndash3

(ii)

20 40 60














INTERACTIONS












(a)0

ndash10

ndash20

ndash30

ndash40

ndash50

ndash60

ndash70

tran

smem

bran

e po

tent

ial (

mV

)

(b)0

ndash10

ndash20

ndash30

ndash40

ndash50

ndash60

ndash70

tran

smem

bran

e po

tent

ial (

mV

)

200 400 600 800 1000

time (ms)

1200 1400 1600 1800 2000







1000

2000

3000

4000

5000

6000

7000

8000

tim

e (m

s)H

z fi

rst

ndash5 0

time-series

5 20 40

time-frequency

60frequency (Hz)

ndash5 0

time-series

5 20 40

time-frequency

60

(a)

(b)

(c)

10

prop

orti

on o

f va

rian

ce

1

06

08

04

02

0

20

30

40

50

60

20 40Hz second

60

(i)

p -va

lue

(unc

orre

cted

)

100

10ndash10

10ndash20

(ii)

20 40 60









1000

2000

3000

4000

5000

6000

7000

8000

tim

e (m

s)H

z fi

rst

ndash5 0

time-series

5 20 40

time-frequency

60frequency (Hz)

ndash5 0

time-series

5 20 40

time-frequency

60

(a)

(b)

(c)

10

prop

orti

on o

f va

rian

ce

08

06

04

02

0

20

30

40

50

60

20 40Hz second

60

(i)

p -va

lue

(unc

orre

cted

)

100

10ndash5

10ndash10

10ndash15(ii)












7 CONCLUSION















Fmax = 165 df 979 p = 0000

(a)

(b)

(c)

100

10ndash5

10ndash10

10ndash15

p -va

lue

(unc

orre

cted

)

0 20 40

frequency (Hz)

60 80

01

0

ndash01

ndash02

ndash03

250

200

150

100

50

tim

e (m

s)

50 100 150 200 250time (ms)

1000 2000 3000 4000time (ms)

5000 6000 7000 8000







(a)

(c)

(d )

10

20

30

40

50

60

freq

uenc

y (H

z)

4

3

1

2

0

10

30

20

50

40

60

freq

uenc

y (H

z)

2000 4000time (ms)

6000 8000

(b)

2000 4000time (ms)

6000 8000

100 200 300 400 500time (ms)

600 700 800 900 1000







Fmax = 45 df 9164 p = 0001

(b)

(a)

(c)

100

10ndash3

10ndash2

10ndash1

10ndash4

10ndash5

p -va

lue

(unc

orre

cted

)

0 20 40

frequency (Hz)

60 80

03

02

01

0

ndash01

250

200

150

100

50

tim

e (m

s)

50 100 150 200 250time (ms)

40002000 6000 8000 10 000time (ms)

12 000 14 000 16 000



AND VOLTERRA SERIES



xi ˆ f3(s)




Onpermilx(t)Š ˆX

j1

X

jn

Z1

0

Z1

0

h nj1jn (u1 un)


















) (A1)

(B1)










Vv ˆ iexcl10 mV





g(t) ˆ js(t)j2


ˆZ














COUPLING







(C1)









let h0 ˆ g0

h1(u1) ˆXp

iˆ 1

g1i bi(u1)

h2(u1u2) ˆXp

iˆ 1

Xp

jˆ 1



y(t) ˆ g0 DaggerXp

iˆ 1

g1i zi(t) Dagger

Xp

iˆ1

Xp

jˆ 1




REFERENCES



















(E2)




































































(a)


(b)


Pjk

excitatory

inhibitory





open closed

channel kinetics

gkt



h0 = 0

03

02

01

0

h1

time [ ms]0 40 80 120

120

80

40

h2

time [ ms]0 40 80 120

(1 ndash gk)Pk



3 NEURONAL CODES











neuronal codes



synchronous(linear)

transient


non-oscillatory






















xTI xj ˆ l1u

T1 v1 Dagger l2u

T2 v2 Dagger l3u

T3 v3 Dagger (4)













g2(0t) ˆX



























2000

4000

6000

8000

10 000

12 000

14 000

16 000

time

(ms)

Hz

firs

t

ndash5 0

time-series

5 20 40

time-frequency

60frequency (Hz)

ndash5 0

time-series

5 20 40

time-frequency

60

(a)

(b)

(c)

10

prop

ortio

n of

var

ianc

e

02

015

01

005

0

20

30

40

50

60

20 40Hz second

60

(i)

p -va

lue

(unc

orre

cted

)

100

10ndash1

10ndash2

10ndash3

(ii)

20 40 60














INTERACTIONS












(a)0

ndash10

ndash20

ndash30

ndash40

ndash50

ndash60

ndash70

tran

smem

bran

e po

tent

ial (

mV

)

(b)0

ndash10

ndash20

ndash30

ndash40

ndash50

ndash60

ndash70

tran

smem

bran

e po

tent

ial (

mV

)

200 400 600 800 1000

time (ms)

1200 1400 1600 1800 2000







1000

2000

3000

4000

5000

6000

7000

8000

tim

e (m

s)H

z fi

rst

ndash5 0

time-series

5 20 40

time-frequency

60frequency (Hz)

ndash5 0

time-series

5 20 40

time-frequency

60

(a)

(b)

(c)

10

prop

orti

on o

f va

rian

ce

1

06

08

04

02

0

20

30

40

50

60

20 40Hz second

60

(i)

p -va

lue

(unc

orre

cted

)

100

10ndash10

10ndash20

(ii)

20 40 60









1000

2000

3000

4000

5000

6000

7000

8000

tim

e (m

s)H

z fi

rst

ndash5 0

time-series

5 20 40

time-frequency

60frequency (Hz)

ndash5 0

time-series

5 20 40

time-frequency

60

(a)

(b)

(c)

10

prop

orti

on o

f va

rian

ce

08

06

04

02

0

20

30

40

50

60

20 40Hz second

60

(i)

p -va

lue

(unc

orre

cted

)

100

10ndash5

10ndash10

10ndash15(ii)












7 CONCLUSION















Fmax = 165 df 979 p = 0000

(a)

(b)

(c)

100

10ndash5

10ndash10

10ndash15

p -va

lue

(unc

orre

cted

)

0 20 40

frequency (Hz)

60 80

01

0

ndash01

ndash02

ndash03

250

200

150

100

50

tim

e (m

s)

50 100 150 200 250time (ms)

1000 2000 3000 4000time (ms)

5000 6000 7000 8000







(a)

(c)

(d )

10

20

30

40

50

60

freq

uenc

y (H

z)

4

3

1

2

0

10

30

20

50

40

60

freq

uenc

y (H

z)

2000 4000time (ms)

6000 8000

(b)

2000 4000time (ms)

6000 8000

100 200 300 400 500time (ms)

600 700 800 900 1000







Fmax = 45 df 9164 p = 0001

(b)

(a)

(c)

100

10ndash3

10ndash2

10ndash1

10ndash4

10ndash5

p -va

lue

(unc

orre

cted

)

0 20 40

frequency (Hz)

60 80

03

02

01

0

ndash01

250

200

150

100

50

tim

e (m

s)

50 100 150 200 250time (ms)

40002000 6000 8000 10 000time (ms)

12 000 14 000 16 000



AND VOLTERRA SERIES



xi ˆ f3(s)




Onpermilx(t)Š ˆX

j1

X

jn

Z1

0

Z1

0

h nj1jn (u1 un)


















) (A1)

(B1)










Vv ˆ iexcl10 mV





g(t) ˆ js(t)j2


ˆZ














COUPLING







(C1)









let h0 ˆ g0

h1(u1) ˆXp

iˆ 1

g1i bi(u1)

h2(u1u2) ˆXp

iˆ 1

Xp

jˆ 1



y(t) ˆ g0 DaggerXp

iˆ 1

g1i zi(t) Dagger

Xp

iˆ1

Xp

jˆ 1




REFERENCES



















(E2)














































3 NEURONAL CODES











neuronal codes



synchronous(linear)

transient


non-oscillatory






















xTI xj ˆ l1u

T1 v1 Dagger l2u

T2 v2 Dagger l3u

T3 v3 Dagger (4)













g2(0t) ˆX



























2000

4000

6000

8000

10 000

12 000

14 000

16 000

time

(ms)

Hz

firs

t

ndash5 0

time-series

5 20 40

time-frequency

60frequency (Hz)

ndash5 0

time-series

5 20 40

time-frequency

60

(a)

(b)

(c)

10

prop

ortio

n of

var

ianc

e

02

015

01

005

0

20

30

40

50

60

20 40Hz second

60

(i)

p -va

lue

(unc

orre

cted

)

100

10ndash1

10ndash2

10ndash3

(ii)

20 40 60














INTERACTIONS












(a)0

ndash10

ndash20

ndash30

ndash40

ndash50

ndash60

ndash70

tran

smem

bran

e po

tent

ial (

mV

)

(b)0

ndash10

ndash20

ndash30

ndash40

ndash50

ndash60

ndash70

tran

smem

bran

e po

tent

ial (

mV

)

200 400 600 800 1000

time (ms)

1200 1400 1600 1800 2000







1000

2000

3000

4000

5000

6000

7000

8000

tim

e (m

s)H

z fi

rst

ndash5 0

time-series

5 20 40

time-frequency

60frequency (Hz)

ndash5 0

time-series

5 20 40

time-frequency

60

(a)

(b)

(c)

10

prop

orti

on o

f va

rian

ce

1

06

08

04

02

0

20

30

40

50

60

20 40Hz second

60

(i)

p -va

lue

(unc

orre

cted

)

100

10ndash10

10ndash20

(ii)

20 40 60









1000

2000

3000

4000

5000

6000

7000

8000

tim

e (m

s)H

z fi

rst

ndash5 0

time-series

5 20 40

time-frequency

60frequency (Hz)

ndash5 0

time-series

5 20 40

time-frequency

60

(a)

(b)

(c)

10

prop

orti

on o

f va

rian

ce

08

06

04

02

0

20

30

40

50

60

20 40Hz second

60

(i)

p -va

lue

(unc

orre

cted

)

100

10ndash5

10ndash10

10ndash15(ii)












7 CONCLUSION















Fmax = 165 df 979 p = 0000

(a)

(b)

(c)

100

10ndash5

10ndash10

10ndash15

p -va

lue

(unc

orre

cted

)

0 20 40

frequency (Hz)

60 80

01

0

ndash01

ndash02

ndash03

250

200

150

100

50

tim

e (m

s)

50 100 150 200 250time (ms)

1000 2000 3000 4000time (ms)

5000 6000 7000 8000







(a)

(c)

(d )

10

20

30

40

50

60

freq

uenc

y (H

z)

4

3

1

2

0

10

30

20

50

40

60

freq

uenc

y (H

z)

2000 4000time (ms)

6000 8000

(b)

2000 4000time (ms)

6000 8000

100 200 300 400 500time (ms)

600 700 800 900 1000







Fmax = 45 df 9164 p = 0001

(b)

(a)

(c)

100

10ndash3

10ndash2

10ndash1

10ndash4

10ndash5

p -va

lue

(unc

orre

cted

)

0 20 40

frequency (Hz)

60 80

03

02

01

0

ndash01

250

200

150

100

50

tim

e (m

s)

50 100 150 200 250time (ms)

40002000 6000 8000 10 000time (ms)

12 000 14 000 16 000



AND VOLTERRA SERIES



xi ˆ f3(s)




Onpermilx(t)Š ˆX

j1

X

jn

Z1

0

Z1

0

h nj1jn (u1 un)


















) (A1)

(B1)










Vv ˆ iexcl10 mV





g(t) ˆ js(t)j2


ˆZ














COUPLING







(C1)









let h0 ˆ g0

h1(u1) ˆXp

iˆ 1

g1i bi(u1)

h2(u1u2) ˆXp

iˆ 1

Xp

jˆ 1



y(t) ˆ g0 DaggerXp

iˆ 1

g1i zi(t) Dagger

Xp

iˆ1

Xp

jˆ 1




REFERENCES



















(E2)

































































xTI xj ˆ l1u

T1 v1 Dagger l2u

T2 v2 Dagger l3u

T3 v3 Dagger (4)













g2(0t) ˆX



























2000

4000

6000

8000

10 000

12 000

14 000

16 000

time

(ms)

Hz

firs

t

ndash5 0

time-series

5 20 40

time-frequency

60frequency (Hz)

ndash5 0

time-series

5 20 40

time-frequency

60

(a)

(b)

(c)

10

prop

ortio

n of

var

ianc

e

02

015

01

005

0

20

30

40

50

60

20 40Hz second

60

(i)

p -va

lue

(unc

orre

cted

)

100

10ndash1

10ndash2

10ndash3

(ii)

20 40 60














INTERACTIONS












(a)0

ndash10

ndash20

ndash30

ndash40

ndash50

ndash60

ndash70

tran

smem

bran

e po

tent

ial (

mV

)

(b)0

ndash10

ndash20

ndash30

ndash40

ndash50

ndash60

ndash70

tran

smem

bran

e po

tent

ial (

mV

)

200 400 600 800 1000

time (ms)

1200 1400 1600 1800 2000







1000

2000

3000

4000

5000

6000

7000

8000

tim

e (m

s)H

z fi

rst

ndash5 0

time-series

5 20 40

time-frequency

60frequency (Hz)

ndash5 0

time-series

5 20 40

time-frequency

60

(a)

(b)

(c)

10

prop

orti

on o

f va

rian

ce

1

06

08

04

02

0

20

30

40

50

60

20 40Hz second

60

(i)

p -va

lue

(unc

orre

cted

)

100

10ndash10

10ndash20

(ii)

20 40 60









1000

2000

3000

4000

5000

6000

7000

8000

tim

e (m

s)H

z fi

rst

ndash5 0

time-series

5 20 40

time-frequency

60frequency (Hz)

ndash5 0

time-series

5 20 40

time-frequency

60

(a)

(b)

(c)

10

prop

orti

on o

f va

rian

ce

08

06

04

02

0

20

30

40

50

60

20 40Hz second

60

(i)

p -va

lue

(unc

orre

cted

)

100

10ndash5

10ndash10

10ndash15(ii)












7 CONCLUSION















Fmax = 165 df 979 p = 0000

(a)

(b)

(c)

100

10ndash5

10ndash10

10ndash15

p -va

lue

(unc

orre

cted

)

0 20 40

frequency (Hz)

60 80

01

0

ndash01

ndash02

ndash03

250

200

150

100

50

tim

e (m

s)

50 100 150 200 250time (ms)

1000 2000 3000 4000time (ms)

5000 6000 7000 8000







(a)

(c)

(d )

10

20

30

40

50

60

freq

uenc

y (H

z)

4

3

1

2

0

10

30

20

50

40

60

freq

uenc

y (H

z)

2000 4000time (ms)

6000 8000

(b)

2000 4000time (ms)

6000 8000

100 200 300 400 500time (ms)

600 700 800 900 1000







Fmax = 45 df 9164 p = 0001

(b)

(a)

(c)

100

10ndash3

10ndash2

10ndash1

10ndash4

10ndash5

p -va

lue

(unc

orre

cted

)

0 20 40

frequency (Hz)

60 80

03

02

01

0

ndash01

250

200

150

100

50

tim

e (m

s)

50 100 150 200 250time (ms)

40002000 6000 8000 10 000time (ms)

12 000 14 000 16 000



AND VOLTERRA SERIES



xi ˆ f3(s)




Onpermilx(t)Š ˆX

j1

X

jn

Z1

0

Z1

0

h nj1jn (u1 un)


















) (A1)

(B1)










Vv ˆ iexcl10 mV





g(t) ˆ js(t)j2


ˆZ














COUPLING







(C1)









let h0 ˆ g0

h1(u1) ˆXp

iˆ 1

g1i bi(u1)

h2(u1u2) ˆXp

iˆ 1

Xp

jˆ 1



y(t) ˆ g0 DaggerXp

iˆ 1

g1i zi(t) Dagger

Xp

iˆ1

Xp

jˆ 1




REFERENCES



















(E2)


















































g2(0t) ˆX



























2000

4000

6000

8000

10 000

12 000

14 000

16 000

time

(ms)

Hz

firs

t

ndash5 0

time-series

5 20 40

time-frequency

60frequency (Hz)

ndash5 0

time-series

5 20 40

time-frequency

60

(a)

(b)

(c)

10

prop

ortio

n of

var

ianc

e

02

015

01

005

0

20

30

40

50

60

20 40Hz second

60

(i)

p -va

lue

(unc

orre

cted

)

100

10ndash1

10ndash2

10ndash3

(ii)

20 40 60














INTERACTIONS












(a)0

ndash10

ndash20

ndash30

ndash40

ndash50

ndash60

ndash70

tran

smem

bran

e po

tent

ial (

mV

)

(b)0

ndash10

ndash20

ndash30

ndash40

ndash50

ndash60

ndash70

tran

smem

bran

e po

tent

ial (

mV

)

200 400 600 800 1000

time (ms)

1200 1400 1600 1800 2000







1000

2000

3000

4000

5000

6000

7000

8000

tim

e (m

s)H

z fi

rst

ndash5 0

time-series

5 20 40

time-frequency

60frequency (Hz)

ndash5 0

time-series

5 20 40

time-frequency

60

(a)

(b)

(c)

10

prop

orti

on o

f va

rian

ce

1

06

08

04

02

0

20

30

40

50

60

20 40Hz second

60

(i)

p -va

lue

(unc

orre

cted

)

100

10ndash10

10ndash20

(ii)

20 40 60









1000

2000

3000

4000

5000

6000

7000

8000

tim

e (m

s)H

z fi

rst

ndash5 0

time-series

5 20 40

time-frequency

60frequency (Hz)

ndash5 0

time-series

5 20 40

time-frequency

60

(a)

(b)

(c)

10

prop

orti

on o

f va

rian

ce

08

06

04

02

0

20

30

40

50

60

20 40Hz second

60

(i)

p -va

lue

(unc

orre

cted

)

100

10ndash5

10ndash10

10ndash15(ii)












7 CONCLUSION















Fmax = 165 df 979 p = 0000

(a)

(b)

(c)

100

10ndash5

10ndash10

10ndash15

p -va

lue

(unc

orre

cted

)

0 20 40

frequency (Hz)

60 80

01

0

ndash01

ndash02

ndash03

250

200

150

100

50

tim

e (m

s)

50 100 150 200 250time (ms)

1000 2000 3000 4000time (ms)

5000 6000 7000 8000







(a)

(c)

(d )

10

20

30

40

50

60

freq

uenc

y (H

z)

4

3

1

2

0

10

30

20

50

40

60

freq

uenc

y (H

z)

2000 4000time (ms)

6000 8000

(b)

2000 4000time (ms)

6000 8000

100 200 300 400 500time (ms)

600 700 800 900 1000







Fmax = 45 df 9164 p = 0001

(b)

(a)

(c)

100

10ndash3

10ndash2

10ndash1

10ndash4

10ndash5

p -va

lue

(unc

orre

cted

)

0 20 40

frequency (Hz)

60 80

03

02

01

0

ndash01

250

200

150

100

50

tim

e (m

s)

50 100 150 200 250time (ms)

40002000 6000 8000 10 000time (ms)

12 000 14 000 16 000



AND VOLTERRA SERIES



xi ˆ f3(s)




Onpermilx(t)Š ˆX

j1

X

jn

Z1

0

Z1

0

h nj1jn (u1 un)


















) (A1)

(B1)










Vv ˆ iexcl10 mV





g(t) ˆ js(t)j2


ˆZ














COUPLING







(C1)









let h0 ˆ g0

h1(u1) ˆXp

iˆ 1

g1i bi(u1)

h2(u1u2) ˆXp

iˆ 1

Xp

jˆ 1



y(t) ˆ g0 DaggerXp

iˆ 1

g1i zi(t) Dagger

Xp

iˆ1

Xp

jˆ 1




REFERENCES



















(E2)































































2000

4000

6000

8000

10 000

12 000

14 000

16 000

time

(ms)

Hz

firs

t

ndash5 0

time-series

5 20 40

time-frequency

60frequency (Hz)

ndash5 0

time-series

5 20 40

time-frequency

60

(a)

(b)

(c)

10

prop

ortio

n of

var

ianc

e

02

015

01

005

0

20

30

40

50

60

20 40Hz second

60

(i)

p -va

lue

(unc

orre

cted

)

100

10ndash1

10ndash2

10ndash3

(ii)

20 40 60














INTERACTIONS












(a)0

ndash10

ndash20

ndash30

ndash40

ndash50

ndash60

ndash70

tran

smem

bran

e po

tent

ial (

mV

)

(b)0

ndash10

ndash20

ndash30

ndash40

ndash50

ndash60

ndash70

tran

smem

bran

e po

tent

ial (

mV

)

200 400 600 800 1000

time (ms)

1200 1400 1600 1800 2000







1000

2000

3000

4000

5000

6000

7000

8000

tim

e (m

s)H

z fi

rst

ndash5 0

time-series

5 20 40

time-frequency

60frequency (Hz)

ndash5 0

time-series

5 20 40

time-frequency

60

(a)

(b)

(c)

10

prop

orti

on o

f va

rian

ce

1

06

08

04

02

0

20

30

40

50

60

20 40Hz second

60

(i)

p -va

lue

(unc

orre

cted

)

100

10ndash10

10ndash20

(ii)

20 40 60









1000

2000

3000

4000

5000

6000

7000

8000

tim

e (m

s)H

z fi

rst

ndash5 0

time-series

5 20 40

time-frequency

60frequency (Hz)

ndash5 0

time-series

5 20 40

time-frequency

60

(a)

(b)

(c)

10

prop

orti

on o

f va

rian

ce

08

06

04

02

0

20

30

40

50

60

20 40Hz second

60

(i)

p -va

lue

(unc

orre

cted

)

100

10ndash5

10ndash10

10ndash15(ii)












7 CONCLUSION















Fmax = 165 df 979 p = 0000

(a)

(b)

(c)

100

10ndash5

10ndash10

10ndash15

p -va

lue

(unc

orre

cted

)

0 20 40

frequency (Hz)

60 80

01

0

ndash01

ndash02

ndash03

250

200

150

100

50

tim

e (m

s)

50 100 150 200 250time (ms)

1000 2000 3000 4000time (ms)

5000 6000 7000 8000







(a)

(c)

(d )

10

20

30

40

50

60

freq

uenc

y (H

z)

4

3

1

2

0

10

30

20

50

40

60

freq

uenc

y (H

z)

2000 4000time (ms)

6000 8000

(b)

2000 4000time (ms)

6000 8000

100 200 300 400 500time (ms)

600 700 800 900 1000







Fmax = 45 df 9164 p = 0001

(b)

(a)

(c)

100

10ndash3

10ndash2

10ndash1

10ndash4

10ndash5

p -va

lue

(unc

orre

cted

)

0 20 40

frequency (Hz)

60 80

03

02

01

0

ndash01

250

200

150

100

50

tim

e (m

s)

50 100 150 200 250time (ms)

40002000 6000 8000 10 000time (ms)

12 000 14 000 16 000



AND VOLTERRA SERIES



xi ˆ f3(s)




Onpermilx(t)Š ˆX

j1

X

jn

Z1

0

Z1

0

h nj1jn (u1 un)


















) (A1)

(B1)










Vv ˆ iexcl10 mV





g(t) ˆ js(t)j2


ˆZ














COUPLING







(C1)









let h0 ˆ g0

h1(u1) ˆXp

iˆ 1

g1i bi(u1)

h2(u1u2) ˆXp

iˆ 1

Xp

jˆ 1



y(t) ˆ g0 DaggerXp

iˆ 1

g1i zi(t) Dagger

Xp

iˆ1

Xp

jˆ 1




REFERENCES



















(E2)
























































INTERACTIONS












(a)0

ndash10

ndash20

ndash30

ndash40

ndash50

ndash60

ndash70

tran

smem

bran

e po

tent

ial (

mV

)

(b)0

ndash10

ndash20

ndash30

ndash40

ndash50

ndash60

ndash70

tran

smem

bran

e po

tent

ial (

mV

)

200 400 600 800 1000

time (ms)

1200 1400 1600 1800 2000







1000

2000

3000

4000

5000

6000

7000

8000

tim

e (m

s)H

z fi

rst

ndash5 0

time-series

5 20 40

time-frequency

60frequency (Hz)

ndash5 0

time-series

5 20 40

time-frequency

60

(a)

(b)

(c)

10

prop

orti

on o

f va

rian

ce

1

06

08

04

02

0

20

30

40

50

60

20 40Hz second

60

(i)

p -va

lue

(unc

orre

cted

)

100

10ndash10

10ndash20

(ii)

20 40 60









1000

2000

3000

4000

5000

6000

7000

8000

tim

e (m

s)H

z fi

rst

ndash5 0

time-series

5 20 40

time-frequency

60frequency (Hz)

ndash5 0

time-series

5 20 40

time-frequency

60

(a)

(b)

(c)

10

prop

orti

on o

f va

rian

ce

08

06

04

02

0

20

30

40

50

60

20 40Hz second

60

(i)

p -va

lue

(unc

orre

cted

)

100

10ndash5

10ndash10

10ndash15(ii)












7 CONCLUSION















Fmax = 165 df 979 p = 0000

(a)

(b)

(c)

100

10ndash5

10ndash10

10ndash15

p -va

lue

(unc

orre

cted

)

0 20 40

frequency (Hz)

60 80

01

0

ndash01

ndash02

ndash03

250

200

150

100

50

tim

e (m

s)

50 100 150 200 250time (ms)

1000 2000 3000 4000time (ms)

5000 6000 7000 8000







(a)

(c)

(d )

10

20

30

40

50

60

freq

uenc

y (H

z)

4

3

1

2

0

10

30

20

50

40

60

freq

uenc

y (H

z)

2000 4000time (ms)

6000 8000

(b)

2000 4000time (ms)

6000 8000

100 200 300 400 500time (ms)

600 700 800 900 1000







Fmax = 45 df 9164 p = 0001

(b)

(a)

(c)

100

10ndash3

10ndash2

10ndash1

10ndash4

10ndash5

p -va

lue

(unc

orre

cted

)

0 20 40

frequency (Hz)

60 80

03

02

01

0

ndash01

250

200

150

100

50

tim

e (m

s)

50 100 150 200 250time (ms)

40002000 6000 8000 10 000time (ms)

12 000 14 000 16 000



AND VOLTERRA SERIES



xi ˆ f3(s)




Onpermilx(t)Š ˆX

j1

X

jn

Z1

0

Z1

0

h nj1jn (u1 un)


















) (A1)

(B1)










Vv ˆ iexcl10 mV





g(t) ˆ js(t)j2


ˆZ














COUPLING







(C1)









let h0 ˆ g0

h1(u1) ˆXp

iˆ 1

g1i bi(u1)

h2(u1u2) ˆXp

iˆ 1

Xp

jˆ 1



y(t) ˆ g0 DaggerXp

iˆ 1

g1i zi(t) Dagger

Xp

iˆ1

Xp

jˆ 1




REFERENCES



















(E2)




















































(a)0

ndash10

ndash20

ndash30

ndash40

ndash50

ndash60

ndash70

tran

smem

bran

e po

tent

ial (

mV

)

(b)0

ndash10

ndash20

ndash30

ndash40

ndash50

ndash60

ndash70

tran

smem

bran

e po

tent

ial (

mV

)

200 400 600 800 1000

time (ms)

1200 1400 1600 1800 2000







1000

2000

3000

4000

5000

6000

7000

8000

tim

e (m

s)H

z fi

rst

ndash5 0

time-series

5 20 40

time-frequency

60frequency (Hz)

ndash5 0

time-series

5 20 40

time-frequency

60

(a)

(b)

(c)

10

prop

orti

on o

f va

rian

ce

1

06

08

04

02

0

20

30

40

50

60

20 40Hz second

60

(i)

p -va

lue

(unc

orre

cted

)

100

10ndash10

10ndash20

(ii)

20 40 60









1000

2000

3000

4000

5000

6000

7000

8000

tim

e (m

s)H

z fi

rst

ndash5 0

time-series

5 20 40

time-frequency

60frequency (Hz)

ndash5 0

time-series

5 20 40

time-frequency

60

(a)

(b)

(c)

10

prop

orti

on o

f va

rian

ce

08

06

04

02

0

20

30

40

50

60

20 40Hz second

60

(i)

p -va

lue

(unc

orre

cted

)

100

10ndash5

10ndash10

10ndash15(ii)












7 CONCLUSION















Fmax = 165 df 979 p = 0000

(a)

(b)

(c)

100

10ndash5

10ndash10

10ndash15

p -va

lue

(unc

orre

cted

)

0 20 40

frequency (Hz)

60 80

01

0

ndash01

ndash02

ndash03

250

200

150

100

50

tim

e (m

s)

50 100 150 200 250time (ms)

1000 2000 3000 4000time (ms)

5000 6000 7000 8000







(a)

(c)

(d )

10

20

30

40

50

60

freq

uenc

y (H

z)

4

3

1

2

0

10

30

20

50

40

60

freq

uenc

y (H

z)

2000 4000time (ms)

6000 8000

(b)

2000 4000time (ms)

6000 8000

100 200 300 400 500time (ms)

600 700 800 900 1000







Fmax = 45 df 9164 p = 0001

(b)

(a)

(c)

100

10ndash3

10ndash2

10ndash1

10ndash4

10ndash5

p -va

lue

(unc

orre

cted

)

0 20 40

frequency (Hz)

60 80

03

02

01

0

ndash01

250

200

150

100

50

tim

e (m

s)

50 100 150 200 250time (ms)

40002000 6000 8000 10 000time (ms)

12 000 14 000 16 000



AND VOLTERRA SERIES



xi ˆ f3(s)




Onpermilx(t)Š ˆX

j1

X

jn

Z1

0

Z1

0

h nj1jn (u1 un)


















) (A1)

(B1)










Vv ˆ iexcl10 mV





g(t) ˆ js(t)j2


ˆZ














COUPLING







(C1)









let h0 ˆ g0

h1(u1) ˆXp

iˆ 1

g1i bi(u1)

h2(u1u2) ˆXp

iˆ 1

Xp

jˆ 1



y(t) ˆ g0 DaggerXp

iˆ 1

g1i zi(t) Dagger

Xp

iˆ1

Xp

jˆ 1




REFERENCES



















(E2)


















































1000

2000

3000

4000

5000

6000

7000

8000

tim

e (m

s)H

z fi

rst

ndash5 0

time-series

5 20 40

time-frequency

60frequency (Hz)

ndash5 0

time-series

5 20 40

time-frequency

60

(a)

(b)

(c)

10

prop

orti

on o

f va

rian

ce

1

06

08

04

02

0

20

30

40

50

60

20 40Hz second

60

(i)

p -va

lue

(unc

orre

cted

)

100

10ndash10

10ndash20

(ii)

20 40 60









1000

2000

3000

4000

5000

6000

7000

8000

tim

e (m

s)H

z fi

rst

ndash5 0

time-series

5 20 40

time-frequency

60frequency (Hz)

ndash5 0

time-series

5 20 40

time-frequency

60

(a)

(b)

(c)

10

prop

orti

on o

f va

rian

ce

08

06

04

02

0

20

30

40

50

60

20 40Hz second

60

(i)

p -va

lue

(unc

orre

cted

)

100

10ndash5

10ndash10

10ndash15(ii)












7 CONCLUSION















Fmax = 165 df 979 p = 0000

(a)

(b)

(c)

100

10ndash5

10ndash10

10ndash15

p -va

lue

(unc

orre

cted

)

0 20 40

frequency (Hz)

60 80

01

0

ndash01

ndash02

ndash03

250

200

150

100

50

tim

e (m

s)

50 100 150 200 250time (ms)

1000 2000 3000 4000time (ms)

5000 6000 7000 8000







(a)

(c)

(d )

10

20

30

40

50

60

freq

uenc

y (H

z)

4

3

1

2

0

10

30

20

50

40

60

freq

uenc

y (H

z)

2000 4000time (ms)

6000 8000

(b)

2000 4000time (ms)

6000 8000

100 200 300 400 500time (ms)

600 700 800 900 1000







Fmax = 45 df 9164 p = 0001

(b)

(a)

(c)

100

10ndash3

10ndash2

10ndash1

10ndash4

10ndash5

p -va

lue

(unc

orre

cted

)

0 20 40

frequency (Hz)

60 80

03

02

01

0

ndash01

250

200

150

100

50

tim

e (m

s)

50 100 150 200 250time (ms)

40002000 6000 8000 10 000time (ms)

12 000 14 000 16 000



AND VOLTERRA SERIES



xi ˆ f3(s)




Onpermilx(t)Š ˆX

j1

X

jn

Z1

0

Z1

0

h nj1jn (u1 un)


















) (A1)

(B1)










Vv ˆ iexcl10 mV





g(t) ˆ js(t)j2


ˆZ














COUPLING







(C1)









let h0 ˆ g0

h1(u1) ˆXp

iˆ 1

g1i bi(u1)

h2(u1u2) ˆXp

iˆ 1

Xp

jˆ 1



y(t) ˆ g0 DaggerXp

iˆ 1

g1i zi(t) Dagger

Xp

iˆ1

Xp

jˆ 1




REFERENCES



















(E2)



















































1000

2000

3000

4000

5000

6000

7000

8000

tim

e (m

s)H

z fi

rst

ndash5 0

time-series

5 20 40

time-frequency

60frequency (Hz)

ndash5 0

time-series

5 20 40

time-frequency

60

(a)

(b)

(c)

10

prop

orti

on o

f va

rian

ce

08

06

04

02

0

20

30

40

50

60

20 40Hz second

60

(i)

p -va

lue

(unc

orre

cted

)

100

10ndash5

10ndash10

10ndash15(ii)












7 CONCLUSION















Fmax = 165 df 979 p = 0000

(a)

(b)

(c)

100

10ndash5

10ndash10

10ndash15

p -va

lue

(unc

orre

cted

)

0 20 40

frequency (Hz)

60 80

01

0

ndash01

ndash02

ndash03

250

200

150

100

50

tim

e (m

s)

50 100 150 200 250time (ms)

1000 2000 3000 4000time (ms)

5000 6000 7000 8000







(a)

(c)

(d )

10

20

30

40

50

60

freq

uenc

y (H

z)

4

3

1

2

0

10

30

20

50

40

60

freq

uenc

y (H

z)

2000 4000time (ms)

6000 8000

(b)

2000 4000time (ms)

6000 8000

100 200 300 400 500time (ms)

600 700 800 900 1000







Fmax = 45 df 9164 p = 0001

(b)

(a)

(c)

100

10ndash3

10ndash2

10ndash1

10ndash4

10ndash5

p -va

lue

(unc

orre

cted

)

0 20 40

frequency (Hz)

60 80

03

02

01

0

ndash01

250

200

150

100

50

tim

e (m

s)

50 100 150 200 250time (ms)

40002000 6000 8000 10 000time (ms)

12 000 14 000 16 000



AND VOLTERRA SERIES



xi ˆ f3(s)




Onpermilx(t)Š ˆX

j1

X

jn

Z1

0

Z1

0

h nj1jn (u1 un)


















) (A1)

(B1)










Vv ˆ iexcl10 mV





g(t) ˆ js(t)j2


ˆZ














COUPLING







(C1)









let h0 ˆ g0

h1(u1) ˆXp

iˆ 1

g1i bi(u1)

h2(u1u2) ˆXp

iˆ 1

Xp

jˆ 1



y(t) ˆ g0 DaggerXp

iˆ 1

g1i zi(t) Dagger

Xp

iˆ1

Xp

jˆ 1




REFERENCES



















(E2)






















































7 CONCLUSION















Fmax = 165 df 979 p = 0000

(a)

(b)

(c)

100

10ndash5

10ndash10

10ndash15

p -va

lue

(unc

orre

cted

)

0 20 40

frequency (Hz)

60 80

01

0

ndash01

ndash02

ndash03

250

200

150

100

50

tim

e (m

s)

50 100 150 200 250time (ms)

1000 2000 3000 4000time (ms)

5000 6000 7000 8000







(a)

(c)

(d )

10

20

30

40

50

60

freq

uenc

y (H

z)

4

3

1

2

0

10

30

20

50

40

60

freq

uenc

y (H

z)

2000 4000time (ms)

6000 8000

(b)

2000 4000time (ms)

6000 8000

100 200 300 400 500time (ms)

600 700 800 900 1000







Fmax = 45 df 9164 p = 0001

(b)

(a)

(c)

100

10ndash3

10ndash2

10ndash1

10ndash4

10ndash5

p -va

lue

(unc

orre

cted

)

0 20 40

frequency (Hz)

60 80

03

02

01

0

ndash01

250

200

150

100

50

tim

e (m

s)

50 100 150 200 250time (ms)

40002000 6000 8000 10 000time (ms)

12 000 14 000 16 000



AND VOLTERRA SERIES



xi ˆ f3(s)




Onpermilx(t)Š ˆX

j1

X

jn

Z1

0

Z1

0

h nj1jn (u1 un)


















) (A1)

(B1)










Vv ˆ iexcl10 mV





g(t) ˆ js(t)j2


ˆZ














COUPLING







(C1)









let h0 ˆ g0

h1(u1) ˆXp

iˆ 1

g1i bi(u1)

h2(u1u2) ˆXp

iˆ 1

Xp

jˆ 1



y(t) ˆ g0 DaggerXp

iˆ 1

g1i zi(t) Dagger

Xp

iˆ1

Xp

jˆ 1




REFERENCES



















(E2)




















































Fmax = 165 df 979 p = 0000

(a)

(b)

(c)

100

10ndash5

10ndash10

10ndash15

p -va

lue

(unc

orre

cted

)

0 20 40

frequency (Hz)

60 80

01

0

ndash01

ndash02

ndash03

250

200

150

100

50

tim

e (m

s)

50 100 150 200 250time (ms)

1000 2000 3000 4000time (ms)

5000 6000 7000 8000







(a)

(c)

(d )

10

20

30

40

50

60

freq

uenc

y (H

z)

4

3

1

2

0

10

30

20

50

40

60

freq

uenc

y (H

z)

2000 4000time (ms)

6000 8000

(b)

2000 4000time (ms)

6000 8000

100 200 300 400 500time (ms)

600 700 800 900 1000







Fmax = 45 df 9164 p = 0001

(b)

(a)

(c)

100

10ndash3

10ndash2

10ndash1

10ndash4

10ndash5

p -va

lue

(unc

orre

cted

)

0 20 40

frequency (Hz)

60 80

03

02

01

0

ndash01

250

200

150

100

50

tim

e (m

s)

50 100 150 200 250time (ms)

40002000 6000 8000 10 000time (ms)

12 000 14 000 16 000



AND VOLTERRA SERIES



xi ˆ f3(s)




Onpermilx(t)Š ˆX

j1

X

jn

Z1

0

Z1

0

h nj1jn (u1 un)


















) (A1)

(B1)










Vv ˆ iexcl10 mV





g(t) ˆ js(t)j2


ˆZ














COUPLING







(C1)









let h0 ˆ g0

h1(u1) ˆXp

iˆ 1

g1i bi(u1)

h2(u1u2) ˆXp

iˆ 1

Xp

jˆ 1



y(t) ˆ g0 DaggerXp

iˆ 1

g1i zi(t) Dagger

Xp

iˆ1

Xp

jˆ 1




REFERENCES



















(E2)


















































(a)

(c)

(d )

10

20

30

40

50

60

freq

uenc

y (H

z)

4

3

1

2

0

10

30

20

50

40

60

freq

uenc

y (H

z)

2000 4000time (ms)

6000 8000

(b)

2000 4000time (ms)

6000 8000

100 200 300 400 500time (ms)

600 700 800 900 1000







Fmax = 45 df 9164 p = 0001

(b)

(a)

(c)

100

10ndash3

10ndash2

10ndash1

10ndash4

10ndash5

p -va

lue

(unc

orre

cted

)

0 20 40

frequency (Hz)

60 80

03

02

01

0

ndash01

250

200

150

100

50

tim

e (m

s)

50 100 150 200 250time (ms)

40002000 6000 8000 10 000time (ms)

12 000 14 000 16 000



AND VOLTERRA SERIES



xi ˆ f3(s)




Onpermilx(t)Š ˆX

j1

X

jn

Z1

0

Z1

0

h nj1jn (u1 un)


















) (A1)

(B1)










Vv ˆ iexcl10 mV





g(t) ˆ js(t)j2


ˆZ














COUPLING







(C1)









let h0 ˆ g0

h1(u1) ˆXp

iˆ 1

g1i bi(u1)

h2(u1u2) ˆXp

iˆ 1

Xp

jˆ 1



y(t) ˆ g0 DaggerXp

iˆ 1

g1i zi(t) Dagger

Xp

iˆ1

Xp

jˆ 1




REFERENCES



















(E2)


















































Fmax = 45 df 9164 p = 0001

(b)

(a)

(c)

100

10ndash3

10ndash2

10ndash1

10ndash4

10ndash5

p -va

lue

(unc

orre

cted

)

0 20 40

frequency (Hz)

60 80

03

02

01

0

ndash01

250

200

150

100

50

tim

e (m

s)

50 100 150 200 250time (ms)

40002000 6000 8000 10 000time (ms)

12 000 14 000 16 000



AND VOLTERRA SERIES



xi ˆ f3(s)




Onpermilx(t)Š ˆX

j1

X

jn

Z1

0

Z1

0

h nj1jn (u1 un)


















) (A1)

(B1)










Vv ˆ iexcl10 mV





g(t) ˆ js(t)j2


ˆZ














COUPLING







(C1)









let h0 ˆ g0

h1(u1) ˆXp

iˆ 1

g1i bi(u1)

h2(u1u2) ˆXp

iˆ 1

Xp

jˆ 1



y(t) ˆ g0 DaggerXp

iˆ 1

g1i zi(t) Dagger

Xp

iˆ1

Xp

jˆ 1




REFERENCES



















(E2)














































AND VOLTERRA SERIES



xi ˆ f3(s)




Onpermilx(t)Š ˆX

j1

X

jn

Z1

0

Z1

0

h nj1jn (u1 un)


















) (A1)

(B1)










Vv ˆ iexcl10 mV





g(t) ˆ js(t)j2


ˆZ














COUPLING







(C1)









let h0 ˆ g0

h1(u1) ˆXp

iˆ 1

g1i bi(u1)

h2(u1u2) ˆXp

iˆ 1

Xp

jˆ 1



y(t) ˆ g0 DaggerXp

iˆ 1

g1i zi(t) Dagger

Xp

iˆ1

Xp

jˆ 1




REFERENCES



















(E2)






















































Vv ˆ iexcl10 mV





g(t) ˆ js(t)j2


ˆZ














COUPLING







(C1)









let h0 ˆ g0

h1(u1) ˆXp

iˆ 1

g1i bi(u1)

h2(u1u2) ˆXp

iˆ 1

Xp

jˆ 1



y(t) ˆ g0 DaggerXp

iˆ 1

g1i zi(t) Dagger

Xp

iˆ1

Xp

jˆ 1




REFERENCES



















(E2)





















































let h0 ˆ g0

h1(u1) ˆXp

iˆ 1

g1i bi(u1)

h2(u1u2) ˆXp

iˆ 1

Xp

jˆ 1



y(t) ˆ g0 DaggerXp

iˆ 1

g1i zi(t) Dagger

Xp

iˆ1

Xp

jˆ 1




REFERENCES



















(E2)













































the labile brain. i. neuronal transients and nonlinear ...karl/the labile brain i.pdf · the labile...

Documents