Cognitive Neurophysiologyof the Motor Cortex

Apostolos P. Georgopoulos,* Masato Taira, Alexander LukashinA major challenge of current neuroscience is to elucidate the brain mechanisms thatunderlie cognitive function. There is no doubt that cognitive processing in the brain engageslarge populations of cells. This article explores the logic of investigating these problemsby combining psychological studies in human subjects and neurophysiological studies ofneuronal populations in the motor cortex of behaving monkeys. The results obtained showthat time-varying psychological processes can be visualized in the time-varying activity ofneuronal populations. Moreover, the functional interactions between cells in the motorcortex are very similar to those observed in a massively interconnected artificial networkperforming the same computation.

The recording of the activity of single cellsin the brain of behaving animals provides atool for directly studying the functionalproperties of single cells, interactions be-tween cells, and the dynamics of neuronalpopulations involved in a variety of cogni-tive processes, including attention, memo-ry, perception, and motor intention. Thismethod was introduced 36 years ago byRicci, Doane, and Jasper (1) and was per-fected and popularized later by Evarts (2).Evarts saw it as the only way to studyvoluntary movement, a function that bydefinition cannot be studied in anesthetizedpreparations. His and subsequent studiesshowed that changes in cell activity in themotor cortex precede the development ofthe motor output and relate quantitativelyto its intensity (3) and spatial character-istics (4-7).

The main challenge with data obtainedwith this technique in studies of cognitivefunction is their interpretation. For exam-ple, a common finding is that cell activitychanges in a certain brain area during aparticular cognitive process. The crucialquestion is: how can we deduce the time-varying cognitive process from the single-cell recordings? How can purely temporalseries of action potentials (spike trains)

A. P. Georgopoulos holds the American Legion andUniversity of Minnesota Brain Sciences Chair in theBrain Sciences Center, Department of Veterans AffairsMedical Center, Minneapolis, MN 55417, and is in theDepartments of Physiology and Neurology, Universityof Minnesota Medical School, Minneapolis, MN 55455.M. Taira is in the Brain Sciences Center, Departmentof Veterans Affairs Medical Center, Minneapolis, MN55417; the Department of Physiology, University ofMinnesota Medical School, Minneapolis, MN 55455;and is on leave of absence from Nihon University,Tokyo, Japan. A. Lukashin is in the Brain SciencesCenter, Department of Veterans Affairs Medical Cen-ter, Minneapolis, MN 55417; the Department of Phys-iology, University of Minnesota Medical School, Min-neapolis, MN 55455; and is on leave of absence fromthe Institute of Molecular Genetics, Russian Academyof Sciences, 123182 Moscow, Russia.

*To whom correspondence should be addressed.

yield information about a cognitive processunfolding in time?

To solve this problem, one must realizethat a cognitive process usually operates ona variable; for example, mental arithmeticoperates on numbers. The process, then, isthe operation, and the puzzle of how thebrain performs multiplication is trans-formed to the problem of how numbers arebeing multiplied. The crucial idea is that ifwe can decipher the neural coding of num-bers, then we have a good chance of deci-phering the cognitive process of mentalmultiplication by observing neural activityduring this process, recovering the numbersby decoding, and inferring how they arebeing operated upon in this particular pro-cess. The essence of the idea is that solvingthe problem of neural coding of a particularvariable provides the means for potentiallysolving the problem of cognitive processingof that variable. These logical steps areshown in Table 1. The crucial step is step 2,namely that of neural coding. This is thestep that connects cell activity with thevariable of interest-that is, the step thatprovides the link between the neural di-mension and the dimension of the variable.We illustrate below the successful applica-tion of this sequence of investigation to thestudy of cognitive processes involving mo-tor operations in space. For that purpose,we chose the direction of reaching move-ment as the spatial variable of interest. We

Table 1. Steps in deciphering brain mecha-nisms of cognitive processes.

1. Select a variable of interest.2. Find the neural coding of the variable

outside the cognitive process.3. Select a cognitive process operating on

the variable of interest.4. Record brain activity during cognitive

processing and infer how the variable isoperated on.

wanted to know how directional informa-tion is encoded in the motor cortex and howcognitive processes operating on direction(for example, memory or mental rotation)are reflected in motor cortical activity.

The Problem of Coding: SingleCells and Neuronal Populations

The problem with the coding of the direc-tion of movement in space is that directionis a closed (circular or spherical) variableand as such does not lend itself to simplemonotonic coding by the intensity of cellactivity. A possible simple solution to theproblem would be to allocate cells thatwould be specifically activated only withmovements in a particular direction-thatis, for cells to be sharply tuned to thedirection of movement. However, this is notthe case. Instead, cells in the motor cortex(4-7) as well as in other structures (8, 9) arebroadly tuned to the direction ofmovement.This means that the cell activity is highestfor a movement in a particular direction (thecell's preferred direction) and decreases pro-gressively with movements farther away fromthis direction. The changes in cell activityrelate to the direction and not the target ofthe reaching movement (10). Quantitative-ly, the crucial variable on which cell activitydepends is the angle formed between thedirection of the movement and the cell'spreferred direction: the intensity of cell ac-tivity can be approximated as a linear func-tion of the cosine of this angle (4-9). Thedirectional tuning equation is

Di(Mk) = bi+aicos OCM^ (1)where Di(Mk) is the discharge rate of the ithcell with movement in direction Mk, bi anda, are regression coefficients, and 0CMk is theangle between the direction of movementMk and the cell's preferred direction Ci. Anexample is shown in Fig. 1. Some pointsconcerning preferred directions are note-worthy. First, cells in a cortical columntend to have very similar preferred direc-tions (11). Second, particular preferreddirections are multiply represented in themotor cortex (11). And third, the pre-ferred directions of single cells are notclustered in particular directions but rangethroughout the directional continuum (4-9) (Fig. 2). This indicates a distributedvectorial coding rather than coding of acoordinate frame (12); for example, ifsuch a frame were Cartesian, the preferreddirections would have clustered along thethree cardinal directions.

The broad directional tuning indicatesthat a given cell participates in movements ofvarious directions; from this result and fromthe fact that preferred directions range widely,it follows that a movement in a particulardirection will involve the engagement of a

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Fig. 1. Three-dimensionaldirectional tuning. The axes(white) meet at the origin ofthe movement. For a partic-ular movement, the dis-charge rate of the cell pre-dicted by Eq. 1 is propor-tional to the length of a linepointing in the direction ofthe movement and drawnfrom the origin to the sur-face (purple) of the tuningvolume. The cell's preferreddirection is indicated by theyellow cone.

whole population of cells. How, then, is thedirection of reaching encoded in an unambig-uous fashion in a population of neurons, eachof which is directionally broadly tuned? Toanswer this question, we hypothesized thatthe motor cortical command for the directionof reaching can be regarded as an ensemble ofvectors (13, 14) in which each vector repre-sents the contribution of a directionally tunedcell. A particular vector points in the cell'spreferred direction and has a length propor-tional to the change in cell activity associatedwith a particular movement direction. For agiven movement Mk, the vector sum of theseweighted cell vectors (the neuronal popula-tion vector P) can be regarded as the outcomeof the ensemble operation

P(Mk) = X Vi(Mk)Ci

where C, is the preferred direction (4) ofthe ith cell and Vi(Mk) is the activity of the

Fig. 2. Three-dimensionalpreferred directions (pur-ple) of 634 motor corticalcells studied in three mon-keys. The axes are in white.

ith cell averaged over a period of time (forexample, the reaction time). The popula-tion vector points at or near the direction ofthe movement (12-16) (Fig. 3). Threeaspects of the population vector are note-worthy: its simplicity, its robustness, and itsspatial characteristics. First, the calculationof the population vector is a simple proce-dure for it (i) assumes directional selectivityof single cells, which is apparent; (ii)weights vectorial contributions by singlecells on the basis of the change in cellactivity, which is reasonable; and (iii) is theoutcome of the vectorial summation ofthese contributions, which is practically thesimplest procedure to obtain a unique out-come. Second, the population vector is arobust measure, for it can still convey agood directional signal even with relativelyfew (100 to 150) cells (15). And third, thepopulation vector is a spatial measure. Thepopulation analysis transforms aggregates of

Fig. 3. The population vector (green) obtainedfrom the set of cells with preferred directionsshown in Fig. 2. The direction of movement isshown in yellow.

purely temporal spike trains into a spatio-temporal population vector.

The neuronal population performing thevectorial operation consists only of direc-tionally tuned cells. Given that the preferreddirection seems to be represented in corticalcolumns (11) and that the population oper-ation involves cells with different preferreddirections, it follows that this operation hasto be intercolumnar. Moreover, the popula-tion vector is a good predictor of the direc-tion of movement when it is calculatedseparately from subsets of cells recorded inthe upper or lower cortical layers (17).

The neuronal population vector hasproved to be a robust and accurate measureof the directional tendency of a neuronalensemble under a variety of conditions, in-cluding movements from different origins (9,18), continuous drawing movements (19),and isometric force pulses (6). Moreover,the analysis holds in other structures con-cemed with sensorimotor control (8, 9) orvisual processing (20). Single cell activity isbroadly tuned in other areas (21) and forother movements (22), although a popula-tion vector analysis has not been performed.Finally, a cosine directional tuning was ob-served in the elements of the hidden layer ofa three-layer artificial network trained to per-form the population vector operation (23).

Especially interesting is a recent general-ization of the application of the populationvector analysis to the coding of faces in thedischarge of cells in the inferotemporal cor-tex of monkeys (24). In the studies of themotor cortex, the population vector and thevectorial contributions of single cells were indirections in physical space. The face codingstudy generalized the vector approach to anarbitrary space of multidimensional scalingof the similarity of face features. Thus, spaceneed not be physical but can be any n-di-mensional feature space. Even for motorfunction, this can be a powerful approach.For example, an interesting question is howthe motor cortex controls finger movementsduring hand manipulation of objects thatinvolve a large number of combinations of

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Problem Human behavior

Problem

Psychological process

New problem

Brain eventsFig. 4. Cognitive neuroscience loop.

finger movements. The hypothesis would bethat (i) single cells code for combinations ofmanipulatory movements, (ii) a particularcell discharges most for a preferred combina-tion, (iii) the intensity of cell activationfollows a broad, possibly cosine, tuning withthe various movement combinations whenthey are expressed in a continuum of simi-larity in a reduced "movement combinationspace," and (iv) when cell contributions areexpressed as weighted vectors in the latterspace, their vector sum (population vector)would provide an unequivocal signal for thecoding of a particular manipulatory move-ment combination. Given a task that canprovide the requisite variety of movementcombinations, the hypothesis above can betested rigorously.

The Population Vector as aTemporal Probe of Direction

The population vector can be used as aprobe by which to monitor in time thechanging directional tendency of the neu-ronal ensemble. One can obtain the timeevolution of the population vector (Eq. 2)by calculating it at short successive inter-vals t (for example, every 10 or 20 ms) orcontinuously, during periods of interest:

P(Mk;t) = 1 Vi(Mi;t)C, (3)

Thus, cortical mechanisms that underliespecific processes (for example, memoriza-tion) could be followed by observation ofthe time-varying population vector.

The feasibility of this approach was firstdocumented when it was shown that theneuronal population vector predicts thedirection of movement during the reactiontime (11, 14). The visual reaction time is aperiod of approximately 300 ms that inter-venes between the appearance of a visualtarget and the initiation of movement; dur-ing this period, the upcoming movement isbeing planned and its execution initiated.This is the simplest case of predicting thedirection of the upcoming movement. Inaddition, the population vector predictswell the direction of movement during aninstructed delay period (25). In these ex-periments, monkeys were trained to with-hold the movement for a period of time

Fig. 5. Rotation of the neuronal populationvector during the reaction time from the direc-tion of the stimulus (blue) to the direction of themovement (red). Data from all eight stimulusdirections used (31) are shown.

after the onset of a visual cue signal and tomove later in response to a "go" signal.During this instructed delay period, thepopulation vector gave a reliable signalconcerning the direction of the movementthat was triggered later for execution. Fi-nally, in an even more complex task, thepopulation vector predicted well the direc-tion ofmovement during a memorized delayperiod (26). In these experiments, the tar-get of the movement was shown for only300 ms. The monkeys were trained towithhold the movement for a subsequentperiod of time, during which the target wasoff, and then moved in the direction of thememorized target in response to a "go"signal. During this memorized delay period,the population vector pointed in the direc-tion of the memorized movement.

Neural Mechanisms of a CognitiveProcess: Mental Rotation

The cognitive process we chose for studyinvolved a transformation of an intendedmovement direction. Our general approachin studying the brain mechanisms of acognitive function involves (i) defining thecognitive task, (ii) performing psychologi-cal experiments in human subjects, theresults of which lead to hypotheses con-cerning the nature of the cognitive process,(iii) training monkeys to perform the sametask and recording the activity of singlecells in the brains of these animals duringperformance of the task, and (iv) connect-ing the neural results with those of thehuman studies and interpreting the psycho-logical results on the basis of the neurophys-iological ones. This cycle is illustrated inFig. 4: the objective is to get as close aspossible to relating neurophysiology andcognitive psychology. Below, we describethese steps as they were applied to a partic-ular problem of a mental transformation of

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movement direction. Subjects were re-quired to move a handle at an angle from areference direction defined by a visual stim-ulus on a plane. Because the referencedirection changed from trial to trial, thetask required that in a given trial thedirection of movement be specified accord-ing to this reference direction.

In human studies, subjects performedblocks of 20 trials in which the angle and itsdeparture (counterclockwise or clockwise)were fixed, although the reference directionvaried (27). Seven angles (50 to 1400) wereused. The basic finding was that the reactiontime increased in a linear fashion with theangle. The most parsimonious hypothesis toexplain this result is that subjects arrive atthe correct direction of movement by shift-ing their motor intention from the referencedirection to the movement direction, trav-eling through the intermediate angularspace. This idea is very similar to the mentalrotation hypothesis advanced by Shepardand co-workers (28) to explain the mono-tonic increase of the reaction time withorientation angle about when a judgmenthas to be made about whether a visual imageis normal or mirror-image. Interestingly, themean rates of rotation (approximately 400°per second) and their range among subjectsare very similar in both kinds of study.When the same human subjects performedboth perceptual and motor rotation tasks,their processing rates were positively corre-lated (29), a result that indicates similarprocessing constraints for both tasks.

The results of neurophysiological studies(30, 31) provided direct evidence for themental rotation hypothesis. Rhesus mon-keys were trained to move the handle 900and counterclockwise from a reference di-rection. The population vector rotated dur-ing the reaction time from the stimulus(reference) direction to the direction of themovement through the counterclockwiseangle. This is illustrated in Fig. 5. Theoccurrence of a true rotation was furtherdocumented by showing that there was atransient increase during the middle of thereaction time in the recruitment of cellswith preferred directions between the stim-ulus and movement directions (31). Thisneural rotation process, sweeping throughthe directionally tuned ensemble, providedfor the first time a direct visualization of adynamic cognitive process (32). The meanrotation rate and the range of rates observedfor different reference directions (31) werevery similar to those obtained in the humanstudies (27, 29).

The Motor Cortex as a Network:Real and Artificial

The population vector and its transforma-tions are the results of operations within an

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ensemble of cells. The cell activity is, inturn, the result of converging influences onthe motor cortex of both external signalsfrom other brain areas and of intrinsicinteractions among the motor cortical cells.We investigated these local interactions byrecording the impulse activity of severalcells simultaneously by using seven inde-pendently movable microelectrodes (33).The data used here come from recordings ofimpulse activity during the reaction andmovement time in the motor cortex of fivemonkeys during the performance of a reach-ing task (4). The seven electrodes werearranged in a linear array every 0.6 mm;cells were recorded at all interelectrodedistances. A total of 1728 pairs were used inthis analysis; in 1126 pairs, both cells in apair were directionally tuned, whereas inthe remaining 602 pairs none of the twocells in a pair were tuned. We wanted toknow whether the prevalence of significantinteractions differed significantly betweenthe tuned and nontuned pairs and whetherthe strength of interaction was correlatedwith the similarity of preferred directions ofcells in a pair. For that purpose, we estimat-ed the strength of presumed interaction(synaptic weight) from the ith to the jthneuron in a pair using an analysis based onwaiting time probability density function(34). An example is illustrated in Fig. 6.

There were two major findings of ouranalysis. First, significant interactions were2.25 times more frequent in the directionallytuned (203 of 1126 cells or 18%) than in thenontuned (48 of602 or 8%) group (P < lo-;chi-square test); significant interactions in thetuned cell group were observed for cells re-corded at all interelectrode distances. Second,the mean synaptic strength (34) was negative-ly correlated with the angle (00 to 180°)between the preferred directions of the twoneurons [correlation coefficient (r) = -0.815;P < 0.0041 (Fig. 7A) throughout the range ofconnections from positive (excitation) to neg-ative (inhibition). This is illustrated in thecover photograph.

The presence of interactions among cellsin the motor cortex has been suggested onmorphological grounds (35) and demon-strated by electrophysiological techniques(36). Our results demonstrate that the di-rectional tuning of motor cortical cells is asignificant factor governing the strength ofinteractions between cells. This finding isqualitatively similar to that observed in thevisual cortex regarding the association ofcells with similar orientation tuning (37).

The importance of directional tuning forthe presence of cell interactions and thedependence of the strength of these interac-tions on the similarity of preferred directionsprovide a fertile ground on which to testhypotheses concerning the organization ofartificial neural networks performing a pop-

50

Fig. 6. Dependence of cell inter-actions on their preferred direc-tions. Two cases illustrate themethods described in (34). Up-per row, a case of cells with sim-ilar preferred directions (arrows).The "difference distribution" andthe CUSUM are positive, whichindicates an excitatory effect. Thestatistical significance of theCUSUM (P < 0.04) is establishedwhen it crosses the upper chanceline. Lower row, a case of cellswith very different preferred direc-tions. The "difference distribution"and the CUSUM are negative,which indicates an inhibitory ef-fect.

ulation operation (23, 38). For that purposewe used a single layer, extensively intercon-nected nonsymmetric network that consist-ed of directionally tuned cells and performedthe calculation of the neuronal populationvector. We wanted to know whether thestrength of interactions between its elementsdepended on the similarity of their preferreddirections, as observed in the motor cortex.The fact that the neuronal population vectorremains stable after an initial growth (11,15) implies that during this steady-state pe-riod the activities Vi (Eq. 3) cease to change,so that dVi/dt = 0 for all i. Activities Vi canbe represented as V, = g(uj), where u. is theinternal state of the ith neuron (u, for exam-ple, might represent the membrane potentialof the neuron averaged over a reasonabletime interval) and g is an activation functionhaving a saturation nonlinearity. Assumingthat the internal state u is a linear functionof inputs received by the neuron from theother neurons in the network, then at astable state (dV./dt = 0) the equality

Vi(M0) = g[ WijVj(Mk)] (4)j

is valid for all neurons (39) and, generally,for all directions M. The weights wiq in Eq.

Excitation

Difference CUSUM

1i~i+0rnS~ j20

Significancetesting

m220 ms

MS

Inhibition

-27 -192

ems

4 can be regarded as synaptic connectionstrengths. If activities Vi at a stable state areknown for a given number of neurons andfor an appropriate number of directions M,Eq. 4 can be used for determining theparameters w11.

Equation 4 shows that the stability ofthe population vector could be ensured byan appropriate set of synaptic connectionstrengths w%. To determine the general fea-tures of the sets of these strengths that wouldensure stability, we searched for parameters wwithin different ranges of possible values Iwi< d (where d is a restriction parameter) andfor different numbers of neurons N in a net-work. In routine calculations, the activationfunction g in Eq. 4 was specified as g(u) =tanh u. A cosine tuning function was chosenfor activities Vf(Mk) in accordance with ex-perimental findings (4): Vf(Mk) = acos 0ik(Eq. 1), where ai is a positive number and Oikis the angle between the preferred direction ofthe ii cell (C) and the direction of upcomingmovement (Mk). Directions C, and Mk wererandomly and uniformly distributed in space,and the values of a were randomly and uni-fornly distributed on the interval [0,1]. Foreach set of N randomly selected, cell-pre-ferred directions and of K randomly selectedmovement directions, we obtained values of

Fig. 7. The dependence of the mean A Bvalue (+ SEM) of synaptic strength on 1.0. Motor cortex Neural networkthe angle between preferred directions l,of neurons involved in the connection.We calculated the mean value of synap- { 0.5tic strength by averaging over synapticIstrengths between neurons, the pre- OO. Iferred directions of which did not differ . \from each other by more than 180. (A)Results from 203 pairs of directionally -0.5.tuned neurons with significant interac- Etions recorded simultaneously in the Zmonkey motor cortex. The total number -1.06 60 120 180 0 6 0 120 180of values averaged for each of ten points Angle between preferred directions (degrs)plotted (from left to right) are 25, 33, 22,24, 16, 21, 15, 15, 18, and 14. (B) Results of simulations for N = 32, K = 32, and d = 0.2. The totalnumber of values averaged for each of ten points plotted (from left to right) are 134, 104, 104, 94,94, 90, 96, 108, 100, and 100.

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the w parameters that ensured the conditionsdescribed in Eq. 4 above by minimizing a costfunction without assuming a symmetry of thematrix w% (40).

Figure 7B shows the result of the calcu-lations for d = 0.2 and N = 32. Thenormalized mean value of synaptic connec-tion strength w is plotted against the anglebetween the preferred directions of inter-connected neurons: these parameters arenegatively correlated (r = -0.984). Ourcalculations for different values of d and Nconfirmed that this correlation is alwaysstrong enough if possible values of synapticconnection strengths are restricted to d -

1/N but practically disappears when therestriction becomes weak (d ~ 1) (41).

These simulation studies showed thatweakly interconnected but correlated neu-rons can ensure the stability of the popula-tion vector. It is likely that convergingexternal inputs initiate the changes in ac-tivity in the motor cortex and contribute tothe ongoing activity of the population.However, such external contributions canbe understood and evaluated properly onlywithin the context of the dynamics of thecortical network itself. Our results showthat the network can by itself support astable process, which leads to the idea thatexternal inputs may act as initiators ormodifiers but need not be the exclusivedeterminants of this intrinsic process.

There are three points worth mention-ing in comparing our data from the neuraland artificial networks. First, the interac-tions between directionally tuned cellspredicted by the network model were con-firmed by the results of the neurophysio-logical studies. Second, the emphasis inour modeling on intrinsic cell interactionsas means for sustaining cell activities with-in the network is warranted by the recentemphasis on intrinsic cortical interactionsas means for amplifying and sustainingcortical excitation (42). And third, theprediction by our model that extensive butweak interactions are sufficient for thestability of a network operation provides areasonable explanation of, and a possiblefunction for, the extensive (35) but weakinteractions observed between corticalcells (43).

Conclusion

Major progress has been made during thepast decade toward determining the func-tional properties of single motor corticalcells with respect to behavior and the un-derstanding of operations by neuronal pop-ulations. This knowledge, combined withan elucidation of the interactions amongcells and rigorous network modeling,should lead to an understanding of how thecortex works and how cognitive operations

are processed in specific brain areas. Alimitation of the single cell recording tech-nique is that it can be usually applied onlyto one restricted brain area at a time. Othertechniques, including positron emission to-mography, can provide a greater picture ofareas of activation in the brain duringperformance of a task. A new major tool isthe oxygen-based functional imaging of thebrain with the use of nuclear magneticresonance (44). This technique is noninva-sive, sensitive, does not require averagingof data from more than one subject, pos-sesses adequate resolution, and has alreadybeen successfully applied to imaging of thehuman motor cortex (45). This methodprovides information complementary tothat obtained by single cell recordings and,together with the latter, can lead to majorinsights in brain function.

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ibid. 92, 139 (1992).27. A. P. Georgopoulos and J. T. Massey, ibid. 65,

361 (1987).28. R. N. Shepard and J. Metzler, Science 171, 701

(1971); R. N. Shepard and L. A. Cooper, MentalImages and Their Transformations (MIT Press,Cambridge, MA, 1982).

29. G. Pellizzer and A. P. Georgopoulos, Exp. BrainRes., in press.

30. A. P. Georgopoulos, J. T. Lurito, M. Petrides, A. B.Schwartz, J. T. Massey, Science 243, 234 (1989).

31. J. T. Lurito, T. Georgakopoulos, A. P. Georgopou-los, Exp. Brain Res. 87, 562 (1991).

32. J. J. Freyd, Psychol. Rev. 94, 427 (1987).33. V. B. Mountcastle, H. J. Reitboeck, G. F. Poggio,

M. A. Steinmetz, J. Neurosci. Methods 36, 77(1991). The implantation of a recording chamberwas performed aseptically under general pento-barbital (28 mg per kilogram of body weight)anesthesia.

34. C. E. Osborn and R. E. Poppele, ibid. 24, 125(1988). See also D. R. Cox, Renewal Theory(Butler & Tanner, Frome, United Kingdom, 1962)and D. R. Cox and P. A. W. Lewis, The StatisticalAnalysis of Series of Events (Chapman & Hall,London, 1966). We estimated the synaptic con-nection strength w',, from the ith to the jth neuronby calculating the "difference distribution" be-tween the observed and randomly shuffled distri-butions of waiting times (mean of 100 shuffles) fora period of 2 to 20 ms. This is essentially theprobability distribution, above chance, of the oc-currence of a spike in the jth train following a spikein the jth train. The statistical significance of thecumulative sum (CUSUM) of the differences wasthen tested [P. Armitage, Sequential Medical Tri-als (Wiley, New York, 1975)], and for significantlydifferent distributions, the peak signed value ofthe CUSUM was taken as the estimate of thestrength of the interaction.

35. S. R. y Cajal, Histologie du Systdme Nerveux,Tome II (Instituto Ramon y Cajal, Madrid, 1955);R. Porter, in Handbook of Physiology, Section 1:The Nervous System, Volume Il: Motor Control,Part 2, J. M. Brookhart, V. B. Mountcastle, V. B.Brooks, S. R. Geiger, Eds. (American Physiologi-cal Society, Bethesda, MD, 1981), pp. 1063-1081; J. DeFelipe, M. Conley, E. G. Jones, J.Neurosci. 6, 3749 (1986).

36. C. Stefanis and H. Jasper, J. Neurophysiol. 27,828 (1964); ibid., p. 855; V. B. Brooks and H.Asanuma, Arch. Ital. Biol. 103, 247 (1965); K.Takahasi, K. Kubota, M. Uno, J. Neurophysiol. 30,22 (1967); H. Asanuma and 1. Rosen, Exp. BrainRes. 16, 507 (1973); J. H. J. Allum, M.-C. Hepp-Reymond, G. Gysin, Brain Res. 231, 325 (1982);W. S. Smith and E. E. Fetz, Soc. Neurosci. Abstr.12, 256 (1986); H. C. Kwan, J. T. Murphy, Y. C.

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Wong, Brain Res. 400, 259 (1987); J. T. Lurito, A.B. Schwartz, M. Petrides, R. E. Kettner, A. P.Georgopoulos, Soc. Neurosci. Abstr. 14, 342(1988).

37. D. Y. Tso, C. D. Gilbert, T. N. Wiesel, J. Neurosci.6,1160 (1986).

38. L. N. Eisenman, J. Keifer, J. C. Houk, in Analysisand Modelling of Neural Systems, F. Eeckman,Ed. (Kluwer Academic, Norwell, MA, 1991), pp.371-376; Y. Burnod et at., J. Neurosci. 12, 1435(1992).

39. Equation 4 is the usual condition for a desiredstate of the neuronal ensemble V.(M) to be anattractor of standard dynamical rules used inneural network theory [S. Amari, IEEE Trans.Syst. Man Cybern. SMC-2, 643 (1972); T. J.Sejnowski, Biol. Cybern. 22, 203 (1976); S.Grossberg and M. Cohen, IEEE Trans. Syst. ManCybern. SMC-13, 815 (1983); J. J. Hopfield,Proc. Nati. Acad. Sci. U.S.A. 81, 3088 (1984); A.Atiya and P. Baldi, Int. J. Neural Syst. 1, 103(1989)].

40. The cost function (F) was the following:K N

F(w) = (1INK) X lacos oikk i

- g(l w,1ajcos ok) (5)

In routine calculations, the number of neurons Nand the number of directions Kwere varied from 8to 64. Initially, all w values were assigned ran-domly (F - 0.5) on the interval 1w < d, where dis the restriction parameter, and then the param-eters w were adjusted to reduce the F functiondown to 10-5. The latter procedure was carriedout for wI < d by means of the simulated anneal-ing algorithm [S. Kirkpatrick, C. D. Gelatt, Jr., M.P. Vecchi, Science 220, 671 (1983)] with anannealing schedule fitted for the present problem.The cost function above was treated as "energy"of the system. Standard Monte Carlo proceduresof changes in the w parameters were used toobtain the Boltzmann distribution over states (thatis, sets of the w parameters) for a given temper-ature." Generally, if the cooling of the system isslow enough (annealing procedure) the approachguarantees the achievement of the global mini-mum of the system at zero "temperature": thismeans that the resulting set of the w parametersprovides the global minimum F(w) = 0 of theenergy function (Eq. 5). We checked additionallythat sets of the w parameters ensuring F < 10-5

indeed yielded stable attractors when standarddynamical equations (39) were used to describethe temporal behavior of the neuronal ensemble.Moreover, we have checked the robustness of theresults in respect to different series of randomnumbers used in the generation of a particular setof preferred directions and during the realizationof the simulated annealing procedure (ten trialsfor each set of N and d values).

41. A. V. Lukashin, M. Taira, A. P. Georgopoulos,unpublished data.

42. K. A. C. Martin, J. Physiol. (London) 440, 735(1991).

43. _, 0. J. Exp. Physiol. 73, 637 (1988).44. S. Ogawa, T.-M. Lee, A. S. Nayak, P. Glynn,

Magn. Reson. Med. 14, 68 (1990); S. Ogawa andT.-M. Lee, ibid. 16,9 (1990); S. Ogawa, T.-M. Lee,A. R. Kay, D. W. Tank, Proc. Nati. Acad. Sci.U.S.A. 87, 9868 (1990).

45. P. A. Bandettini, E. C. Wong, R. S. Hinks, R. S.Tikofsky, J. S. Hyde, Magn. Reson. Med. 25, 390(1992); S.-G. Kim et al., J. Neurophysiol. 69, 297(1993).

46. Supported by U.S. Public Health Service grantsNS17413 and PSMH48185, Office of Naval Re-search contract N0001 4-88-K-0751, and a grantfrom the Human Frontier Science Program.

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