letters to nature ... . Self-similarity of complex networks Chaoming Song1 , Shlomo Havlin2 & Hernan A. Makse1 Levich Institute and Physics Department , City College of New York , New York , New York 10031 , USA 2 Minerva Center and Department of Physics , Bar-Ilan University , Ramat Gan 52900 , Israel ... 1 This result comes as a surprise , because the exponential increase in equation -LRB- 1 -RRB- has led to the general understanding that complex networks are not self-similar , since self-similarity requires a powerlaw relation between N and l . How can we reconcile the exponential increase in equation -LRB- 1 -RRB- with self-similarity , or -LRB- in other words -RRB- an underlying length-scaleinvariant topology ? At the root of the self-similar properties that we unravel in this study is a scale-invariant renormalization procedure that we show to be valid for dissimilar complex networks . To demonstrate this concept we first consider a self-similar Complex networks have been studied extensively owing to their relevance to many real systems such as the world-wide web , the Internet , energy landscapes and biological and social networks1 5 . A large number of real networks are referred to as ` scale-free ' because they show a power-law distribution of the number of links per node1 ,6,7 . However , it is widely believed that complex networks are not invariant or self-similar under a length-scale transformation . This conclusion originates from the ` smallworld ' property of these networks , which implies that the number of nodes increases exponentially with the ` diameter ' of the network8 11 , rather than the power-law relation expected for a self-similar structure . Here we analyse a variety of real complex networks and find that , on the contrary , they consist of selfrepeating patterns on all length scales . This result is achieved by the application of a renormalization procedure that coarsegrains the system into boxes containing nodes within a given ` size ' . We identify a power-law relation between the number of boxes needed to cover the network and the size of the box , defining a finite self-similar exponent . These fundamental properties help to explain the scale-free nature of complex networks and suggest a common self-organization dynamics . Two fundamental properties of real complex networks have attracted much attention recently : the small-world and the scalefree properties . Many naturally occurring networks are ` small world ' because we can reach a given node from another one , following the path with the smallest number of links between the nodes , in a very small number of steps . This corresponds to the so-called ` six degrees of separation ' in social networks10 . It is mathematically expressed by the slow -LRB- logarithmic -RRB- increase of the average diameter of the network , l ; with the total number of nodes N , l < lnN ; where l is the shortest distance between two nodes and defines the distance metric in complex networks6 ,8,9,11 . Equivalently , we obtain : N < el = l0 1 where l0 is a characteristic length . A second fundamental property in the study of complex networks arises with the discovery that the probability distribution of the number of links per node , P -LRB- k -RRB- -LRB- also known as the degree distribution -RRB- , can be represented by a power-law -LRB- ` scale-free ' -RRB- with a degree exponent g that is usually in the range 2 , g , 3 -LRB- ref . 6 -RRB- : P k < k2g 2 These discoveries have been confirmed in many empirical studies of diverse networks1 4,6,7 . With the aim of providing a deeper understanding of the underlying mechanism that leads to these common features , we need to probe the patterns within the network structure in more detail . The question of connectivity between groups of interconnected nodes on different length scales has received less attention . But many examples exhibit the importance of collective behaviour , such as interactions between communities within social networks , links between clusters of websites of similar subjects , and the highly modular manner in which molecules interact to keep a cell alive . Here we show that real complex networks , such as the world-wide web -LRB- WWW -RRB- , social , protein protein interaction networks -LRB- PIN -RRB- and cellular networks are invariant or self-similar under a lengthscale transformation . 392 Figure 1 The renormalization procedure applied to complex networks . a , Demonstration of the method for different lB . The first column depicts the original network . We tile the system with boxes of size lB -LRB- different colours correspond to different boxes -RRB- . All nodes in a box are connected by a minimum distance smaller than the given lB . For instance , in the case of lB 2 , we identify four boxes that contain the nodes depicted with colours red , orange , white and blue , each containing 3 , 2 , 1 and 2 nodes , respectively . Then we replace each box by a single node ; two renormalized nodes are connected if there is at least one link between the unrenormalized boxes . Thus we obtain the network shown in the second column . The resulting number of boxes needed to tile the network , N B -LRB- lB -RRB- , is plotted in Fig. 2 versus lB to obtain d B as in equation -LRB- 3 -RRB- . The renormalization procedure is applied again and repeated until the network is reduced to a single node -LRB- third and fourth columns for different lB -RRB- . b , The stages in the renormalization scheme applied to the entire WWW . We fix the box size to lB 3 and apply the renormalization for four stages . This corresponds , for instance , to the sequence for the network demonstration depicted in the second row in panel a . We colour the nodes in the web according to the boxes to which they belong . The network is invariant under this renormalization , as explained in the legend of Fig. 2d and the Supplementary Information . NATURE | VOL 433 | 27 JANUARY 2005 | www.nature.com/nature 2005 Nature Publishing Group letters to nature network embedded in euclidean space , of which a classical example would be a fractal percolation cluster at criticality12 . To unfold the self-similar properties of such clusters we calculate the fractal dimension using a ` box-counting ' method and a ` cluster-growing ' method13 . In the first method we cover the percolation cluster with N B boxes of linear size lB . The fractal dimension or box dimension d B is then given by14 : N B < l2dB B 3 In the second method , the network is not covered with boxes . Instead one seed node is chosen at random and a cluster of nodes centred at the seed and separated by a minimum distance l is calculated . The procedure is then repeated by choosing many seed nodes at random and the average ` mass ' of the resulting clusters -LRB- kM cl , defined as the number of nodes in the cluster -RRB- is calculated as a function of l to obtain the following scaling : kM c l < ldf 14 4 defining the fractal cluster dimension d f . Comparing equations -LRB- 4 -RRB- and -LRB- 1 -RRB- implies that d f 1 for complex small-world networks . For a homogeneous network characterized by a narrow degree distribution -LRB- such as a fractal percolation cluster -RRB- the box-counting method of equation -LRB- 3 -RRB- and the cluster-growing method of equation -LRB- 4 -RRB- are equivalent , because every node typically has the same number of links or neighbours . Equation -LRB- 4 -RRB- can then be derived from equation -LRB- 3 -RRB- and d B d f , and this relation has been regularly used . The crux of the matter is to understand how we can calculate a self-similar exponent -LRB- analogous to the fractal dimension in euclidean space -RRB- in complex inhomogeneous networks with a broad degree distribution such as equation -LRB- 2 -RRB- . Under these conditions equation -LRB- 3 -RRB- and -LRB- 4 -RRB- are not equivalent , as will be shown below . The application of the proper covering procedure in the box-counting method -LRB- equation -LRB- 3 -RRB- -RRB- for complex networks unveils a set of selfsimilar properties such as a finite self-similar exponent and a new set of critical exponents for the scale-invariant topology . Figure 1a illustrates the box-covering method using a schematic network composed of eight nodes . For each value of the box size lB , we search for the number of boxes needed to tile the entire network such that each box contains nodes separated by a distance l , lB . This procedure is applied to several different real networks : -LRB- 1 -RRB- a part of the WWW composed of 325,729 web pages that are connected if there is a URL link from one page to another6 -LRB- http://www.nd.edu/,networks -RRB- ; -LRB- 2 -RRB- a social network where the nodes are 392,340 actors linked if they were cast together in at least one film15 ; -LRB- 3 -RRB- the biological networks of protein protein interactions found in Escherichia coli -LRB- 429 proteins -RRB- and Homo sapiens -LRB- 946 proteins -RRB- linked if there is a physical binding between them -LRB- database available via the Database of Interacting Proteins16 ,17 , other PINs are discussed in the Supplementary Information -RRB- , and Figure 2 Self-similar scaling in complex networks . a , The upper panel shows a log-log plot of N B versus lB , revealing the self-similarity of the WWW and actor network according to equation -LRB- 3 -RRB- . The lower panel shows the scaling of s -LRB- lB -RRB- versus lB according to equation -LRB- 9 -RRB- . The error bars are of the order of the symbol size . b , Same as a but for two PINs : H. sapiens and E. coli . Results are analogous to b but with different scaling exponents . c , Same as a for the cellular networks of A. fulgidus , E. coli and C. elegans . d , Invariance of the degree distribution of the WWW under the renormalization for different NATURE | VOL 433 | 27 JANUARY 2005 | www.nature.com/nature box sizes , lB . We show the data collapse of the degree distributions , demonstrating the self-similarity at different scales . The inset shows the scaling of k 0 s -LRB- lB -RRB- k for different lB , whence we obtain the scaling factor s -LRB- lB -RRB- . Moreover , we also apply the renormalization for a fixed box size , for instance lB 3 as shown in Fig. 1b for the WWW , until the network is reduced to a few nodes , and find that P -LRB- k -RRB- is invariant under these multiple renormalizations as well , for several iterations -LRB- see Supplementary Information -RRB- . 393 2005 Nature Publishing Group letters to nature -LRB- 4 -RRB- the cellular networks compiled by ref . 18 using a graphtheoretical representation of all the biochemical pathways based on the WIT integrated-pathway genome database19 -LRB- http://igweb . integratedgenomics.com/IGwit -RRB- of 43 species from Archaea , Bacteria and Eukarya . Here we show the results for Archaeoglobus fulgidus , E. coli and Caenorhabditis elegans18 ; the full database is analysed in the Supplementary Information . It has been previously determined that the WWW and actor networks are small-world and scale-free , characterized by equation -LRB- 2 -RRB- with g 2.6 and 2.2 , respectively1 . For the PINs of E. coli and H. sapiens we find g 2.2 and 2.1 , respectively . All cellular networks are scale-free with average exponent g 2.2 -LRB- ref . 18 -RRB- . We confirm these values and show the results for the WWW in Fig. 2 . Figure 2a and b shows the results of N B -LRB- lB -RRB- according to equation -LRB- 3 -RRB- . They reveal the existence of self-similarity in the WWW , actors and E. coli and H. sapiens PINs with self-similar exponents d B 4.1 , d B 6.3 , and d B 2.3 and d B 2.3 , respectively . The cellular networks are shown in Fig. 2c and have d B 3.5 . We now elaborate on the apparent contradiction between the two definitions of self-similar exponents in complex networks . After performing a renormalization at a given lB , we calculate the mean mass of the boxes covering the network , kM B -LRB- lB -RRB- l , to obtain : kM B lB l ; N = N B lB < ldB B 5 which is corroborated by direct measurements for all the networks , and shown in Fig. 3a for the WWW . On the other hand , the average obtained from the clustergrowing method -LRB- for this calculation we average over single boxes without tiling the system -RRB- gives rise to an exponential growth of the mass : kM c lB l < elB = l1 6 with l1 < 0.78 in accordance with the small-world effect equation -LRB- 1 -RRB- , as seen in Fig. 3a . The topology of scale-free networks is dominated by several highly connected hubs -- the nodes with the largest degree -- implying that most of the nodes are connected to the hubs via one or very few steps . Therefore the average obtained in the cluster-growing method is biased ; the hubs are overrepresented in equation -LRB- 6 -RRB- because almost every node is a neighbour of a hub . By choosing the seed of the clusters at random , there is a very large probability of including the hubs in the clusters . On the other hand , the boxcovering method is a global tiling of the system , providing a flat average over all the nodes : that is , each part of the network is covered with an equal probability . Once a hub -LRB- or any node -RRB- is covered , it can not be covered again . We conclude that equations -LRB- 3 -RRB- and -LRB- 4 -RRB- are not equivalent for inhomogeneous networks with topologies dominated by hubs with a large degree . The biased sampling of the randomly chosen nodes is clearly demonstrated in Fig. 3b . We find that the probability distribution of the mass of the boxes for a given lB is very broad and can be approximated by a power-law : P M B < M 22:2 in the case of the B WWW and lB 4 . On the other hand , the probability distribution of M c is very narrow and can be fitted by a log-normal distribution -LRB- see Fig. 3b -RRB- . In the box-covering method there are many boxes with very large and very small masses , in contrast to the peaked distribution in the cluster-growing method , thus showing the biased nature of the latter method in inhomogeneous networks . This biased average leads to the exponential growth of the mass in equation -LRB- 6 -RRB- and it also explains why the average distance is logarithmic with N , as in equation -LRB- 1 -RRB- . The box-counting method provides a powerful tool for further investigations of network properties because it enables a renormalization procedure , revealing that the self-similar properties and the scale-free degree distribution persist irrespectively of the amount of coarse-graining of the network . Subsequent to the first step of assigning the nodes to the boxes we create a new renormalized network by replacing each box by a single node . Two boxes are then connected , provided that there was at least one link between their constituent nodes . The second column of the panels in Fig. 1a shows this step in the renormalization procedure for the schematic network , while Fig. 1b shows the results for the same procedure applied to the entire WWW for lB 3 . The renormalized network gives rise to a new probability distribution of links , P -LRB- k 0 -RRB- , which is invariant under the renormalization : P k ! P k < k 2g 0 0 7 Figure 2d supports the validity of this scale transformation by showing a data collapse of all distributions with the same g according to equation -LRB- 7 -RRB- for the WWW . Further insight arises from relating the scale-invariant properties -LRB- equation -LRB- 3 -RRB- -RRB- to the scale-free degree distribution -LRB- equation -LRB- 2 -RRB- -RRB- . Plotting -LRB- see inset in Fig. 2d for the WWW -RRB- the number of links k 0 of each node in the renormalized network versus the maximum number of links k in each box of the unrenormalized network exhibits a scaling law : Figure 3 Different averaging techniques lead to qualitatively different results . a , Mean value of the box mass in the box-counting method , kM Bl , and the cluster mass in the cluster growing method , kM cl , for the WWW . The solid lines represent the power-law fit for kM Bl and the exponential fit for kM cl according to equations -LRB- 5 -RRB- and -LRB- 6 -RRB- , respectively . b , Probability distribution of M B and M c for lB 4 for the WWW . The curves are fitted by a power-law and a log-normal distribution , respectively . 394 k ! k s lB k 0 8 This equation defines the scaling transformation in the connectivity distribution . Empirically we find that the scaling factor s -LRB- ,1 -RRB- scales with lB with a new exponent d k : s lB < l2dk B 9 2005 Nature Publishing Group NATURE | VOL 433 | 27 JANUARY 2005 | www.nature.com/nature letters to nature as shown in Fig. 2a for the WWWand actor networks -LRB- with d k 2.5 and d k 5.3 , respectively -RRB- , in Fig. 2b for the protein networks -LRB- d k 2.1 for E. coli and d k 2.2 for H. sapiens -RRB- and in Fig. 2c for the cellular networks with d k 3.2 . Equations -LRB- 8 -RRB- and -LRB- 9 -RRB- shed light on how families of hierarchical sizes are linked together . The larger the families , the fewer links exist . Surprisingly , the same power-law relation exists for large and small families represented by equation -LRB- 2 -RRB- . From equation -LRB- 7 -RRB- we obtain n -LRB- k -RRB- dk n 0 -LRB- k 0 -RRB- dk 0 , where n -LRB- k -RRB- NP -LRB- k -RRB- is the number of nodes with links k and n 0 k 0 N 0 P k 0 is the number of nodes with links k 0 after the renormalization -LRB- N 0 is the total number of nodes in the renormalized network -RRB- . Using equation -LRB- 8 -RRB- , we obtain n -LRB- k -RRB- s 12gn 0 -LRB- k -RRB- . Then , upon renormalizing a network with N total nodes we obtain a smaller number of nodes N 0 according to N 0 s g21N . The total number of nodes in the renormalized network is the number of boxes needed to cover the unrenormalized network at any given lB , so we have N 0 N B -LRB- lB -RRB- . Then , from equations -LRB- 3 -RRB- and -LRB- 9 -RRB- we obtain the relation between the three indexes : g 1 dB = dk 10 ... . Strong polarization enhancement in asymmetric three-component ferroelectric superlattices Ho Nyung Lee , Hans M. Christen , Matthew F. Chisholm , Christopher M. Rouleau & Douglas H. Lowndes Condensed Matter Sciences Division , Oak Ridge National Laboratory , Oak Ridge , Tennessee 37831 , USA ... . Equation -LRB- 10 -RRB- is confirmed for all the networks analysed here -LRB- see Supplementary Information -RRB- . In all cases the calculation of d B and d k and equation -LRB- 10 -RRB- gives rise to the same g exponent as that obtained in the direct calculation of the degree distribution . The significance of this result is that the scale-free properties characterized by g can be related to a more fundamental length-scale invariant property , characterized by the two new indexes d B and A d k. Received 4 August ; accepted 30 November 2004 ; doi : 10.1038 / nature03248 . 1 . Albert , R. & Barabasi , A.-L . Statistical mechanics of complex networks . Rev. Mod . Phys . 74 , 47 97 -LRB- 2002 -RRB- . 2 . Dorogovtsev , S. N. & Mendes , J. F. F. Evolution of Networks : From Biological Nets to the Internet and the WWW -LRB- Oxford Univ. . Press , Oxford , 2003 -RRB- . 3 . Pastor-Satorras , R. & Vespignani , A. Evolution and Structure of the Internet : a Statistical Physics Approach -LRB- Cambridge Univ. . Press , Cambridge , 2004 -RRB- . 4 . Newman , M. E. J . The structure and function of complex networks . SIAM Rev. 45 , 167 256 -LRB- 2003 -RRB- . 5 . Amaral , L. A. N. & Ottino , J. M. Complex networks -- augmenting the framework for the study of complex systems . Eur . Phys . J. B 38 , 147 162 -LRB- 2004 -RRB- . 6 . Albert , R. , Jeong , H. & Barabasi , A.-L . Diameter of the World Wide Web . Nature 401 , 130 131 -LRB- 1999 -RRB- . 7 . Faloutsos , M. , Faloutsos , P. & Faloutsos , C. On power-law relationships of the Internet topology . Comput . Commun . Rev. 29 , 251 262 -LRB- 1999 -RRB- . 8 . Erdos , P. & Renyi , A. On the evolution of random graphs . Publ . Math . Inst . Hung . Acad . Sci . 5 , 17 61 -LRB- 1960 -RRB- . 9 . Bollobas , B. Random Graphs -LRB- Academic , London , 1985 -RRB- . 10 . Milgram , S . The small-world problem . Psychol . Today 2 , 60 67 -LRB- 1967 -RRB- . 11 . Watts , D. J. & Strogatz , S. H. Collective dynamics of ` small-world ' networks . Nature 393 , 440 442 -LRB- 1998 -RRB- . 12 . Bunde , A. & Havlin , S. Fractals and Disordered Systems Ch . 2 -LRB- eds Bunde , A. & Havlin , S. -RRB- 2nd edn -LRB- Springer , Heidelberg , 1996 -RRB- . 13 . Vicsek , T. Fractal Growth Phenomena 2nd edn , Part IV -LRB- World Scientific , Singapore , 1992 -RRB- . 14 . Feder , J. Fractals -LRB- Plenum , New York , 1988 -RRB- . 15 . Barabasi , A.-L . & Albert , R. Emergence of scaling in random networks . Science 286 , 509 512 -LRB- 1999 -RRB- . 16 . Xenarios , I. et al. . DIP : the database of interacting proteins . Nucleic Acids Res . 28 , 289 291 -LRB- 2000 -RRB- . 17 . Database of Interacting Proteins -LRB- DIP -RRB- khttp : / / dip.doe-mbi . ucla.edul -LRB- 2000 -RRB- . 18 . Jeong , H. , Tombor , B. , Albert , R. , Oltvai , Z. N. & Barabasi , A.-L . The large-scale organization of metabolic networks . Nature 407 , 651 654 -LRB- 2000 -RRB- . 19 . Overbeek , R. et al. . WIT : integrated system for high-throughput genome sequence analysis and metabolic reconstruction . Nucleic Acid Res . 28 , 123 125 -LRB- 2000 -RRB- . Supplementary Information accompanies the paper on www.nature.com/nature . Acknowledgements We are grateful to J. Brujic for many discussions . This work is supported by the National Science Foundation , Materials Theory . S.H. thanks the Israel Science Foundation and ONR for support . Competing interests statement The authors declare that they have no competing financial interests . Correspondence and requests for materials should be addressed to H.A.M. -LRB- makse@mailaps.org -RRB- . NATURE | VOL 433 | 27 JANUARY 2005 | www.nature.com/nature Theoretical predictions -- motivated by recent advances in epitaxial engineering -- indicate a wealth of complex behaviour arising in superlattices of perovskite-type metal oxides . These include the enhancement of polarization by strain1 ,2 and the possibility of asymmetric properties in three-component superlattices3 . Here we fabricate superlattices consisting of barium titanate -LRB- BaTiO3 -RRB- , strontium titanate -LRB- SrTiO3 -RRB- and calcium titanate -LRB- CaTiO3 -RRB- with atomic-scale control by high-pressure pulsed laser deposition on conducting , atomically flat strontium ruthenate -LRB- SrRuO3 -RRB- layers . The strain in BaTiO3 layers is fully maintained as long as the BaTiO3 thickness does not exceed the combined thicknesses of the CaTiO3 and SrTiO3 layers . By preserving full strain and combining heterointerfacial couplings , we find an overall 50 % enhancement of the superlattice global polarization with respect to similarly grown pure BaTiO3 , despite the fact that half the layers in the superlattice are nominally nonferroelectric . We further show that even superlattices containing only single-unit-cell layers of BaTiO3 in a paraelectric matrix remain ferroelectric . Our data reveal that the specific interface structure and local asymmetries play an unexpected role in the polarization enhancement . Oxide heterostructures with atomically abrupt interfaces , defined by atomically flat surface terraces and single-unit-cell steps , can now be grown on well-prepared single-stepped substrates4 7 . This advance has encouraged theoretical investigations that have led to predictions of new artificial materials1 3,8 10 . The atomic-scale control of the combining of dissimilar materials is expected to produce striking property enhancements as well as new combinations of desired properties . Here we discuss the experimental realization of one of these predictions , the strain enhancement of ferroelectric polarization . The challenge associated with fabricating such strained structures -- the deliberate and controlled deposition of up to hundreds of individual layers -- remains a formidable task , for which the principal technique used has been high-vacuum molecular beam epitaxy5 ,11 . However , many insulators do not yield the correct oxide stoichiometry -LRB- or expected resulting physical properties -RRB- when grown by molecular beam epitaxy . Furthermore , a shortage of electrically conducting oxide substrates and our stilllimited understanding of the stability and growth mechanisms of conducting-film electrodes have hindered the electrical characterization of oxide superlattices . To address these challenges , we have recently shown that atomically flat , electrically conducting SrRuO3 electrodes can be grown with a surface quality that mimics that of the substrate -LRB- Fig. 1a -RRB- 7 . Pulsed laser deposition -LRB- PLD -RRB- has long been regarded as an effective method for synthesizing various oxide heterostructures12 15 , but obtaining atomically sharp interfaces has been difficult in the comparatively high-pressure processes needed to maintain oxygen stoichiometry . Here we demonstrate the growth by a high-pressure PLD technique of hundreds of individual perovskite layers of BaTiO3 , SrTiO3 and CaTiO3 . These superlattices were grown with layer-by-layer control , yielding as-grown samples with compositionally abrupt interfaces , atomically smooth surfaces , and excellent ferroelectric behaviour that indicated oxygen stoichiometry . 395 2005 Nature Publishing Group

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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articles Neural correlates of decision variables in parietal cortex Michael L. Platt & Paul W. Glimcher Center for Neural Science , New York University , New York , New York 10003 , USA . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . Decision theory proposes that humans and animals decide what to do in a given situation by assessing the relative value of each possible response . This assessment can be computed , in part , from the probability that each action will result in a gain and the magnitude of the gain expected . Here we show that the gain -LRB- or reward -RRB- a monkey can expect to realize from an eye-movement response modulates the activity of neurons in the lateral intraparietal area , an area of primate cortex that is thought to transform visual signals into eye-movement commands . We also show that the activity of these neurons is sensitive to the probability that a particular response will result in a gain . When animals can choose freely between two alternative responses , the choices subjects make and neuronal activation in this area are both correlated with the relative amount of gain that the animal can expect from each response . Our data indicate that a decision-theoretic model may provide a powerful new framework for studying the neural processes that intervene between sensation and action . Neurobiologists have begun to focus increasingly on the study of sensory motor processing , but many of the models used to describe these processes remain rooted in the classic re ex , initially described by Descartes1 and later advocated by Sherrington2 as a model system for studying the connection between sensation and movement . Sherrington , in particular , viewed response selection as a physical link between independent , anatomically distinct sensory and motor systems2 . Until recently , the sensory motor re ex was tacitly accepted by many physiologists as an appropriate model for describing the neural processes that underlie complex behaviour3 5 . More recent ndings indicate that , at least in some cases , the neural events that connect sensation and movement may involve processes other than classical re exive mechanisms6 9 and these data support theoretical approaches that have challenged re ex models directly10 13 . Researchers outside physiology have long argued for richer models of the sensory motor process14 18 . These models invoke the explicit representation of a class of decision variables , which carry information about the environment , are extracted in advance of response selection , aid in the interpretation or processing of sensory data and are a prerequisite for rational decision making19 . Here we describe a formal economic mathematical approach for the physiological study of the sensory motor process , or decision-making , in the lateral intra-parietal area -LRB- LIP -RRB- of the macaque brain . Theoretical treatments of decision making Although mathematical approaches to decision-making propose different formulae for identifying the response a subject should choose under a given set of conditions , nearly all theories require the decision-maker to have some knowledge of two environmental variables : the gain expected to result from an action and the probability that the expected gain will be realized . The expected value theory of Arnaud and Nichole20 , for example , proposes that a rational decision-maker should multiply expected gain by the probability of gain to establish an expected value for each course of action , and then should choose the option with the highest expected value . Although subsequent theorists have proposed different mathematical combinatorial rules that provide more accurate models of the decision-making process , knowledge of the gain expected from a response and the probability of realizing that gain is still considered to be critical to the computation of rational choice19 ,20,22 . Ecological biologists17 ,18 and psychologists14 16 have also developed formal models of decision-making . Most of these models NATURE | VOL 400 | 15 JULY 1999 | www.nature.com assume that animals make decisions that either maximize the rate of energy intake from different options or match their rates of behavioural responding to the reinforcement rates obtained from different choices . These models propose that animals choose on the basis of several variables , including the probability of reinforcement23 and the magnitude of reinforcement24 ,25 associated with different responses . Experimental evidence indicates that expected gain and probability of gain do in uence the choices animals make . However , neurobiological models of the processes that connect sensation and action almost never propose the explicit representation of decision variables by the nervous system . Most contemporary neurophysiological models , particularly models of cortical sensory motor transformations , identify sensory and motor signals as critical components , but they rarely address the possibility that neural signals may encode data that fall outside these two categories . Our investigations have led us to propose that decision theory might provide a powerful alternative framework for studying the sensory motor process . Our current model could be described as having two classes of inputs : current sensory data , and a stored representation of environmental contingencies . Current sensory data would re ect the observer 's best estimate of the current state of the salient elements of the environment . As such , it would be in uenced by stored information that could improve the ef ciency of sensory processing through selective attention . The second class of inputs would represent the chooser 's assumptions about current environmental contingencies , a class of inputs that detail how an action affects the chooser . Decision making would involve the combination of post-attentional sensory data with the subject 's best estimate of the outcome of any given action . These estimates of environmental contingencies would then be combined with a loss function specifying the value of all possible losses and gains to the chooser , according to a decision rule like the one originally proposed by Arnauld and Nichole20 or in subsequent re nements of their model19 ,21,22 . Physiology of sensory motor integration in LIP Although neurobiologists have not employed decision-theoretic formulations of this type , many groups have concluded that a number of brain areas are involved in the processes that intervene between sensation and action . Parietal area LIP has been labelled attentional26 ,27 , decision-related28 and motor-preparatory29 31 by different groups . Our own work indicates that decision-theoretic models may provide a powerful alternative approach to the study of sensory motor processing in parietal cortex32 ,33 . 233 1999 Macmillan Magazines Ltd articles In one study34 , we assumed that visual attention can be treated as conceptually separable from other elements of the decision-making process . We used two tasks that independently controlled both the location of a saccadic target and the location and behavioural relevance of a visual distractor . In both tasks , a monkey initially xated a central stimulus , and then two eccentric stimuli were illuminated , one above and one below the central stimulus . A change in the colour of the central stimulus identi ed one eccentric stimulus as the eventual saccadic target and the other as a visual distractor . In one task , the distractor was highly relevant because a 200 Firing rate -LRB- Hz -RRB- 150 CH990322 * * * ** * * * * * * offset of this stimulus signalled the animal to initiate a saccade to the target ; in the second task the distractor was completely irrelevant , because offset of the central stimulus signalled the animal to initiate the saccade . Although intraparietal neurons were more active on blocks of trials in which a stimulus served as a saccadic target than on blocks of trials in which the same stimulus served as a visual distractor , none of the neuronal responses distinguished between behaviourally relevant and irrelevant distractors . We took this nding as preliminary evidence that activity in area LIP does not participate in sensory-attentional processing but is correlated with either outcome contingencies , gain functions , decision outputs or motor planning , all processes that our decision-theoretic approach indicates are critical in choice behaviour . Experimental design 100 50 0 500 Target onset 500 Early visual Late visual Early cue 1,000 Late cue 1,500 ms PrePostmovement movement b Firing rate -LRB- Hz -RRB- 200 150 100 50 n = 329 0 0.0 0.5 1.0 CH990322 Expected gain ratio c Regression slope 24 18 12 6 0 EV LV EC LC Time Time Here we examine whether a decision-theoretic model might reveal a link between the activity of area LIP neurons and the choices animals make in an oculomotor task . In the rst experiment -LRB- experiment 1 ; Figs 1 , 2 -RRB- , we monitored the activity of single intraparietal neurons while animals performed cued saccade trials , in which a change in the colour of a centrally located xation stimulus instructed subjects to make one of two possible eyemovement responses in order to receive a juice reward . Before this stimulus changed colour , the rewarded movement was ambiguous . In successive blocks of trials , we varied either the volume of juice delivered for each instructed response -LRB- expected gain -RRB- or the probability that each possible response would be instructed -LRB- outcome probability -RRB- , while holding all sensory signals and motorrelated variables constant . This allowed us to test whether any portion of the variation in area LIP spike rates observed under these conditions was correlated with variations in the decision variables we manipulated . In the second experiment -LRB- experiment 2 ; Figs 3 , 4 -RRB- , animals were rewarded for choosing either of two possible eye-movement responses . In sequential blocks of trials , we varied the gain that could be expected from each possible response and then used the frequency with which the animal chose each response as an estimate of the subjective value of each option . This permitted us to test whether neural activity in area LIP was correlated with a behaviourally derived estimate of response value . Our data indicate that some of the variation in the LIP activity was , in fact , correlated with economic decision variables and that these variables in uenced the choices made by our animal subjects and neural activity in a similar manner . Decision variables and LIP spike rates PR PO Figure 1 Modulation of neuronal activity by expected gain . The volume of juice delivered for each response was varied across blocks of cued saccade trials . a , Firing rate of an intraparietal neuron on trials instructing gaze shifts into the response eld , averaged in 100 ms bins and synchronized , at vertical line , on target onset . Thick black line , expected gain ratio 0:75 -LRB- n 48 -RRB- ; thick grey line , expected gain ratio 0:25 -LRB- n 37 -RRB- . Raster panels show spike times during the rst 20 high-gain -LRB- dark grey -RRB- and rst 20 low gain -LRB- light grey -RRB- trials . Panels show individual trials in sequential order from top to bottom . Arrows plot , successively , mean times of instruction cue onset , central xation stimulus offset and saccade onset during high - -LRB- black arrow -RRB- and low - -LRB- grey arrow -RRB- gain blocks . Stars indicate 100-ms bins in which ring rate was signi cantly different in the high-gain block than in the low-gain block -LRB- t-test , P , 0:05 -RRB- . b , Mean ring rate -LRB- 6 s.e. -RRB- during each interval plotted against expected gain ratio , for all trials instructing gaze shifts into the neuronal response eld . Lines indicate best - t linear regressions through the raw data . c , Mean -LRB- 6 s.e. -RRB- regression slope for expected gain -LRB- circles -RRB- and the instructed movement -LRB- triangles -RRB- , plotted against time -LRB- n 40 neurons -RRB- . Intervals : EV , early visual ; LV , late visual ; EC , early cue ; LC , late cue ; PR , pre-movement ; PO , post-movement . Figure 1a shows the activity of an intraparietal neuron on visually identical cued saccade trials during which the monkey was instructed to shift gaze into the neuronal response eld for a juice reward . When , in one block of trials -LRB- thick black line -RRB- , 0.26 ml of juice was delivered , this neuron was more active than when the same movement was made in response to the same visual display but during a block -LRB- thick grey line -RRB- in which only 0.09 ml of juice was delivered . This neuron was studied while expected gain was systematically varied across seven blocks of trials , and behavioural performance was maintained at 91 % correct . To quantify the effects of these manipulations on the activity of this neuron , we computed average ring rates in six 200-ms epochs during each trial -LRB- early visual : 0 200 ms after the onset of the two response targets ; late visual : 200 0 ms before the central light-emitting diode -LRB- LED -RRB- changed colour ; early cue : 0 200 ms after the central LED identi ed the movement that would be reinforced ; late cue : 200 0 ms before the command to initiate the movement ; pre-movement : 0 200 ms before saccade onset ; post-movement : 0 200 ms after saccade onset -RRB- . Figure 1b shows the ring rate of this neuron -LRB- mean 6 s : e :-RRB- as a function of the gain -LRB- in ml of juice -RRB- expected for a movement into the response eld , divided by the sum of the NATURE | VOL 400 | 15 JULY 1999 | www.nature.com 234 1999 Macmillan Magazines Ltd articles Table 1 Percentages of recorded neurons showing signi cant correlations between ring rate and expected gain , outcome probability or estimated value Decision variable Expected gain Outcome probability Estimated value ... . Early visual / xation * 27.5 % -LRB- 17.5 % -RRB- 35 % -LRB- 10 % -RRB- 28 % -LRB- 6 % -RRB- Late visual / xation * 32.5 % -LRB- 15 % -RRB- 35 % -LRB- 10 % -RRB- 17 % -LRB- 28 % -RRB- Early cue/visual * 40 % -LRB- 27.5 % -RRB- 40 % -LRB- 40 % -RRB- 28 % -LRB- 47 % -RRB- Late cue/visual * 27.5 % -LRB- 62.5 % -RRB- 15 % -LRB- 100 % -RRB- 19 % -LRB- 67 % -RRB- Pre-movement 30 % -LRB- 72.5 % -RRB- 30 % -LRB- 95 % -RRB- 19 % -LRB- 67 % -RRB- Post-movement 15 % -LRB- 65 % -RRB- 15 % -LRB- 95 % -RRB- 25 % -LRB- 69 % -RRB- Total ... . Each entry indicates the percentage of cells in each experiment showing signi cant -LRB- P , 0:05 -RRB- modulations in ring rate during each interval . Entries in parentheses indicate the percentage of cells that showed signi cant modulations in ring rate by the movement actually made by the animal . Total percentages indicate the percentage of neurons showing a signi cant modulation in ring rate during any interval . Asterisks identify the names of intervals measured in experiment 2 -LRB- early xation , late xation , early visual , late visual -RRB- . 62.5 % -LRB- 87.5 % -RRB- 75 % -LRB- 100 % -RRB- 56 % -LRB- 89 % -RRB- gain -LRB- in ml -RRB- available from both possible movements , for all those trials in which the animal shifted gaze to the target inside the response eld of the neuron . The ring rate of this intraparietal neuron was correlated with this expected gain ratio early in the trials -LRB- early visual , r 0:432 , P , 0:0001 ; late visual , r 0:372 , a 100 75 50 25 0 500 Target onset 500 1,000 1,500 ms CH990215 ** * * * * * b Firing rate -LRB- Hz -RRB- 100 75 50 25 0 0.0 Early visual Late visual Early cue Late cue PrePostmovement movement n = 363 0.5 1.0 CH990215 Outcome probability Time c 24 18 12 6 0 n = 20 EV LV EC LC Time PR PO Figure 2 Modulation of neuronal activity by outcome probability . The probability that each response would be instructed was varied across blocks of cued saccade trials . a , Firing rate of an intraparietal neuron on trials instructing a gaze shift into the response eld , as in Fig. 1 . Thick black line , outcome probability 0:80 -LRB- n 77 -RRB- ; thick grey line , outcome probability 0:20 -LRB- n 26 -RRB- . b , Mean ring rate -LRB- 6 s.e. -RRB- during each interval plotted against outcome probability , for all trials instructing gaze shifts into the neuronal response eld . c , Mean -LRB- 6 s.e. -RRB- regression slope for outcome probability -LRB- circles -RRB- and the instructed movement -LRB- triangles -RRB- plotted against time -LRB- n 20 neurons -RRB- . NATURE | VOL 400 | 15 JULY 1999 | www.nature.com P , 0:0001 ; early cue , r 0:217 , P , 0:0001 -RRB- , but later in the trials , around the time of the movement , ring rate and expected gain ratio were uncorrelated . These data indicate a temporal shift in the information encoded by this neuron , from expected gain early in trials to the movement made by the animal at the end of trials . On these same trials , movement amplitude -LRB- r 2 0:055 , P . 0:322 -RRB- , movement latency -LRB- r 0:016 , P . 0:767 -RRB- and movement velocity -LRB- r 0:02 , P . 0:718 -RRB- were uncorrelated with expected gain . Each of 40 intraparietal neurons recorded from the brains of three monkeys was tested in 3 7 blocks of trials -LRB- mean 4:7 -RRB- in which the gain associated with each of the two target LEDs was different . For each neuron , correct trials were sorted into two groups based on whether the cued movement was into or out of the response eld . For all trials in which the subject shifted gaze towards the response eld target , we analysed the relationship between ring rates computed during each interval described above and four variables : expected gain ; the amplitude of each saccade ; the average velocity of each saccade ; and the latency of each saccade . We used a multiple regression analysis to determine which portion of ring-rate modulation could be uniquely attributed to changes in expected gain independently of changes in any of the three remaining movementrelated variables . For comparison , a second multiple regression analysis was performed to determine the degree to which the ring rate of each neuron was uniquely in uenced by which movement the subject was cued to make , after correcting for the effects of expected gain , movement amplitude , movement latency and movement velocity . The slopes generated by the multiple regression analyses were used as an index of how strongly changes in expected gain modulated the activation of intraparietal neurons , independently of changes in movement amplitude , latency and velocity . Figure 1c shows the mean regression slope -LRB- 6 s.e. -RRB- for 40 intraparietal neurons for expected gain -LRB- circles -RRB- ; the mean slopes for the instructed movement -LRB- triangles -RRB- are provided for comparison . The regression slopes for expected gain were signi cantly greater than zero -LRB- single-sample t-test , P , 0:05 -RRB- during the early visual and late visual intervals ; expected gain strongly modulated the activation of the population of intraparietal neurons early in trials , before the identi cation of the rewarded movement . After the rewarded movement was identi ed , however , the activation of the intraparietal population was strongly modulated by the instructed movement , but not by expected gain . Table 1 shows the percentages of neurons in this sample that were signi cantly modulated by expected gain during each interval , with the percentages of neurons signi cantly modulated by the instructed movement in each interval provided in parentheses for comparison . Overall , 62.5 % of parietal neurons carried statistically signi cant information about expected gain at some point during the trial , and 87.5 % of neurons carried statistically signi cant information about which movement was actually made . These data indicate that the activation of the intraparietal neuronal population is correlated with expected gain ratio , particularly during periods of uncertainty early in trials when decision variables might be important in formulating a movement plan . 235 Regression slope Firing rate -LRB- Hz -RRB- 1999 Macmillan Magazines Ltd articles Figure 2a shows the modulation of the activity of an intraparietal neuron by changes in outcome probability during visually identical cued saccade trials . When a saccade into the response eld was instructed with a probability of 0.8 -LRB- thick black line -RRB- , the neuron was more active than when an equivalent movement was instructed with a probability of 0.2 -LRB- thick grey line -RRB- . This neuron was studied while outcome probability was varied across seven blocks of trials , and behavioural performance was maintained at 85 % correct . Figure 2b shows the ring rate of this neuron -LRB- mean 6 s : e :-RRB- as a function of outcome probability during each of the six measured epochs , for all trials on which the animal was instructed to shift gaze to the response eld of the neuron . During the early visual -LRB- r 0:407 , P , 0:0001 -RRB- , late visual -LRB- r 0:349 , P , 0:0001 -RRB- , early cue -LRB- r 0:242 , P , 0:0001 -RRB- , late cue -LRB- r 0:131 , P , 0:012 -RRB- and pre-movement intervals -LRB- r 0:116 , P , 0:027 -RRB- , ring rate increased with increases in outcome probability . Movement amplitude -LRB- r 0:056 , P . 0:291 -RRB- , latency -LRB- r 2 0:098 , P . 0:061 -RRB- and velocity -LRB- r 0:07 , P . 0:183 -RRB- were not correlated with outcome probability . The activity of this neuron was thus correlated with the probability that the animal would be instructed to choose each response , independent of the movement actually made by the animal . We studied the effects of varying outcome probability across 5 7 blocks of trials -LRB- mean 6:7 -RRB- on 20 intraparietal neurons in two monkeys . Overall , 75 % of the intraparietal neurons in our sample carried signi cant information about outcome probability at some point during the trial , while all 20 neurons carried signi cant information about which movement was actually made -LRB- Table 1 -RRB- . Figure 2c shows the mean -LRB- 6 s.e. -RRB- regression slopes for outcome probability -LRB- lled circles -RRB- and the instructed movement -LRB- lled triangles -RRB- for our population . The regression slopes for outcome probability were signi cantly greater than zero during the early visual , late visual and early cue intervals -LRB- single-sample t-tests , P , 0:05 -RRB- . The movement made by the animal signi cantly modulated neuronal activity in this population only after the cue identi ed the reinforced movement -LRB- late cue , pre-movement and post-movement intervals , single-sample t-tests , P , 0:05 -RRB- . The modulations induced by changes in outcome probability were very similar to those evoked by changes in expected gain and support the idea that outcome probability in uences decision processes before the colour of the xation stimulus identi es the reinforced movement . The results of experiment 1 show that changes in either the gain expected from a particular response or the probability that a particular response will be required modulate the activity of intraparietal neurons . This is true even when all the visual and motor events associated with each trial are the same . In experiment 1 , however , we were limited in our ability to assess the effects of expected gain and outcome probability on decisionmaking because there was a single correct choice identi ed in each trial , making it dif cult to correlate any aspect of the choices made by our animal subjects with neuronal activity . In experiment 2 , we therefore omitted the centrally located colour cue that identi ed only one of the responses as reinforced and instead allowed subjects to choose either of the two responses while we systematically varied the gain that could be expected from each response across 5 7 blocks of trials -LRB- mean ; 5:8 -RRB- . The actual choices made by subjects were then used as an estimate of the valuation of each response by the animal on each trial and neuronal data was related directly to this behavioural readout of the animal 's decision process . Figure 3a shows , for visually identical trials in which the animal chose to shift gaze into the response eld of the neuron under study , the modulation in activity of an intraparietal neuron in this freechoice task . The neuron was more active during a block of trials rewarded with 0.15 ml of juice -LRB- Fig. 3a , thick black line -RRB- than in 236 a 40 * * *** * Firing rate -LRB- Hz -RRB- 20 CH980304 0 500 Target onset 500 b Firing rate -LRB- Hz -RRB- 40 20 0 0.0 0.5 1.0 Expected gain ratio n = 333 1,000 1,500 ms PrePostmovement movement Early Late fixation fixation Early visual Late visual CH980304 Time Figure 3 Modulation of the activity of a single intraparietal neuron by expected gain in a free-choice task . The volume of juice delivered for each response was varied across blocks of saccadic choice trials . a , Firing rate of an intraparietal neuron on trials in which subjects chose to shift gaze into the response eld , as in Fig. 1 . Thick black line , expected gain ratio 0:75 -LRB- n 75 -RRB- ; thick grey line , expected gain ratio 0:25 -LRB- n 11 -RRB- . Raster panels plot spike times during the rst 10 trials in both the high-gain -LRB- dark grey -RRB- and low-gain -LRB- light grey -RRB- blocks . Arrows plot , sequentially , the mean times of central xation stimulus offset and saccade onset in high - -LRB- black arrows -RRB- and low - -LRB- grey arrows -RRB- gain blocks . b , Mean ring rate -LRB- 6 s.e. -RRB- during each interval plotted against expected gain , for movements into the response eld . Response value and LIP neural activity another block of trials in which the same movement was rewarded with only 0.05 ml of juice -LRB- Fig. 3a , thick grey line -RRB- . In fact , during the high-gain block the neuron became activated before the onset of the two choice targets , a pattern of activation that can not be attributed to a visual response to target onset . We also measured the average ring rate of this neuron during six 200-ms epochs -LRB- early xation , late xation , early visual , late visual , pre-movement and post-movement -RRB- . Figure 3b shows the ring rate of this neuron -LRB- mean 6 s : e :-RRB- during these epochs -LRB- note that two of these intervals , early xation and late xation , are before the illumination of the visual targets -RRB- . Firing rate increased with expected gain during the early xation -LRB- r 0:158 , P , 0:025 -RRB- , late xation -LRB- r 0:181 , P , 0:01 -RRB- , early visual -LRB- r 0:263 , P , 0:001 -RRB- and late visual -LRB- r 0:187 , P , 0:01 -RRB- intervals , but not around the time of the movement . These data con rm the correlation between neuronal activation and the gain that could be expected from a particular movement , as reported in the previous experiments . The goal of this experiment , however , was to directly correlate neuronal activity with the animal 's estimate of the value of the two possible movements . Figure 4a presents the choices the subject made across all blocks of trials during this recording session . Consistent with Herrnstein 's matching law14 for choice behaviour , there was a linear relationship between the proportion of trials on which the animal chose the target inside the response eld and the proportion of total juice available for gaze shifts to that target -LRB- r 0:96 , P , 0:008 -RRB- 35,36 . Across 36 data sets , there was a high correlation between the frequency with which monkeys shifted gaze to a target and the proportion of total juice available for gaze shifts to that target -LRB- mean r 0:89 , mean P 0:03 , mean slope 1:86 , mean intercept 2 0:45 -RRB- . Thus , the choice behaviour of our subjects reliably re ected the gain NATURE | VOL 400 | 15 JULY 1999 | www.nature.com 1999 Macmillan Magazines Ltd articles a Proportion of trials into RF 1.00 Regression slope 0.75 0.50 0.25 0.00 0.00 c 24 18 12 6 0 0.25 0.50 0.75 Expected gain ratio Late fixation 1.00 n = 36 xation and early visual epochs -LRB- single-sample t-tests , P , 0:05 -RRB- . The movement chosen by the animal signi cantly modulated activity in this same population of neurons during the late xation , early visual , late visual and pre-movement epochs . Thus , when an animal is presented with a choice between two possible saccadic gaze shifts , the animal 's estimate of the relative value of these two movements is correlated with the activation of intraparietal neurons . The activation of intraparietal cortex therefore appears to re ect the decision processes animals use to guide behaviour . Discussion EF LF EV LV PR PO Time Late visual PrePostmovement movement b Firing rate -LRB- Hz -RRB- 40 20 Early fixation Early visual 0 0.12 0.0 0.12 n = 333 CH980304 Estimated value Time Figure 4 Correlation of ring rate with a behavioural estimate of the subjective value of the two movements . a , Proportion of trials on which the monkey chose the target inside the neuronal response eld plotted against expected gain . Line indicates best - t linear regression . b , Mean ring rate -LRB- 6 s.e. -RRB- during each interval plotted against a behavioural estimate of the subjective value of each response , computed on each trial from the difference in the reinforcement rates obtained from the two possible movements during the preceding 10 trials24 . c , Mean -LRB- 6 s.e. -RRB- regression slope for estimated value -LRB- lled circles -RRB- and the chosen movement -LRB- lled triangles -RRB- plotted against time -LRB- n 36 neurons -RRB- . Intervals as in Fig. 1c , except : EF , early xation ; LF , late xation . associated with each movement . In subsequent analyses , we therefore used choice frequencies as an estimate of the subject 's relative valuation of the two reinforced responses . To analyse the relationship between the trial-by-trial activity of this neuron and the valuation of each choice by the subject , on each trial we computed a behavioural estimate of the subjective value of a movement into the response eld , based on Herrnstein 's melioration theory23 , by computing the difference in the rate of reinforcement the animal had obtained from each of the two possible choices over the preceding 10 trials -LRB- estimated value -RRB- . Figure 4b shows the mean ring rate of the neuron as a function of this estimated value , during each measured interval , for all trials on which the animal shifted gaze into the response eld . The ring rate of this neuron increased as the estimated value of a movement into the response eld increased during the early xation -LRB- r 0:204 , P , 0:004 -RRB- , late xation -LRB- r 0:152 , P , 0:031 -RRB- , early visual -LRB- r 0:180 , P , 0:01 -RRB- and pre-movement -LRB- r 0:144 , P , 0:04 -RRB- intervals . The ring rate of this neuron was uncorrelated with saccade amplitude , velocity and latency during all six measured intervals , and thus these motor variables can not account for the relationship between ring rate and the subjective value of a movement into the response eld . We recorded the activity of 36 intraparietal neurons from two monkeys performing this free-choice task . Overall , the regression slopes for estimated value were signi cantly greater than zero , at some point during trials , for 56 % of the neurons in the parietal population , while the regression slopes for the chosen movement were signi cantly greater than zero for 89 % of the population -LRB- Table 1 -RRB- . Figure 4c shows the mean -LRB- 6 s.e. -RRB- regression slope for the estimated value of a movement into the response eld -LRB- circles -RRB- during each epoch , as well as the mean -LRB- 6 s.e. -RRB- regression slope for the chosen movement -LRB- triangles -RRB- . The estimated value of a movement into the response eld signi cantly modulated activity in our intraparietal neuronal population during the early xation , late NATURE | VOL 400 | 15 JULY 1999 | www.nature.com Our results indicate that a decision-theoretic model of the sensory motor process may provide a powerful alternative approach to traditional sensory and motor neurophysiological models of response selection . Experiment 1 indicates that both the gain expected from a particular response and the probability that a particular response will be required systematically increase the activation of neurons in posterior parietal cortex . Furthermore , this in uence is separable from the effects of the immediate visual environment and from the neural events that govern movement dynamics . The effects of expected gain and outcome probability were strongest before the time at which the animal knew which of the two possible responses would be rewarded . Experiment 2 shows that when an animal is free to choose between alternative responses , the gain expected from each possible action exerts a correlated in uence on both the choice behaviour of the animal and the activation of posterior parietal neurons . In our free-choice task , both monkeys and posterior parietal neurons behaved as if they had knowledge of the gains associated with different actions . These ndings support the hypothesis that the variables that have been identi ed by economists , psychologists and ecologists as important in decision-making are represented in the nervous system . Our data are , however , dif cult to reconcile with traditional sensory-attentional or motor-physiological models . For example , sensory-attentional models typically attribute modulations in the activity of visually responsive neurons to changes in the behavioural relevance of visual stimuli26 and , based on these models , it could be argued that our manipulations of expected gain and outcome probably merely altered the visual activity of intraparietal neurons . This interpretation seems unlikely for two reasons . First , when an animal is instructed to plan a gaze shift to one of two simultaneously presented eccentric visual stimuli , intraparietal neurons are insensitive to changes in the behavioural relevance of either stimulus when it is not the target of a saccade34 . Second , in experiment 2 we showed that intraparietal neuronal activity was modulated by changes in the estimated value of each response during the xation intervals before the onset of the response targets , when subjects were xating the central LED in otherwise total darkness . We nd this pattern of activation dif cult to attribute to an attentional modulation of visually evoked activity . It might also be suggested that the modulations in neuronal activity we evoked by changes in expected gain , outcome probability and estimated value re ect changes in the degree to which animals plan particular movements . Choice behaviour , such as that shown by subjects in experiment 2 , might thus re ect the outcome of a covert competition between movement plans weighted by the estimated value of each possible action . However , because such a model must include an estimate of response value it is essentially equivalent to a decision-theoretic model . Although the decision variables we have examined here systematically modulated the activity of intraparietal neurons in our tasks , these effects were , for most neurons , con ned to the early portions of each trial . Late in each trial , intraparietal neurons were more strongly modulated by the animal 's movement plan than by expected gain , outcome probability or the animal 's estimate of response value . This may indicate that intraparietal cortex lies close to the motor output stages of the sensory decision motor process . 237 1999 Macmillan Magazines Ltd articles These data may also indicate that other brain areas believed to participate in decision-making , such as prefrontal cortex37 , might represent decision variables throughout the trial . In conclusion , our results indicate that a neurobiological framework for the study of choice based upon the same classical decision theory that has guided behavioural studies of the choices humans and animals make may provide a powerful paradigm for studying the sensory decision motor process . Moreover , this framework indicates that many of the modulatory in uences attributed to sensory attention or motor planning might be more precisely described as a re ection of decision processes . The in uence of these variables on the activity of posterior parietal cortex , a brain area believed to participate in the transformation of sensory signals into motor commands , indicates that a decision-theoretic framework might be a useful tool for understanding the neurophysiology of sensory-guided behaviour . M ... . illumination of a yellow LED located directly in front of the animal . 500 ms after the subject had aligned his gaze -LRB- 62 8 -RRB- with this LED , the two eccentric yellow response LEDs identi ed by our mapping procedure were illuminated . After a random interval -LRB- 500 800 ms -RRB- , the central LED was extinguished and the monkey was rewarded if he shifted gaze into alignment with -LRB- 64 8 -RRB- either eccentric LED . The gain that could be expected from each movement was varied systematically across 5 7 blocks of about 100 trials each . A block in which each movement was reinforced equally was typically run rst , and then pairs of blocks of complementary high gain/low gain conditions were randomly interleaved . Received 30 April ; accepted 25 May 1999 . 1 . Descartes , R. Treatise on Man -LRB- trans . Hall , T. S. -RRB- -LRB- Harvard Univ. . Press , Cambridge , Massachusetts , 1662:1972 -RRB- . 2 . Sherrington , C . 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Sci . 39 , 326 -LRB- 1998 -RRB- . 34 . Platt , M. L. & Glimcher , P. W. Responses of intraparietal neurons to saccadic targets and visual distractors . J. Neurophysiol . 78 , 1574 1589 -LRB- 1997 -RRB- . 35 . Catania , A. C. Concurrent performances : a baseline for the study of reinforcement magnitude . J. Exp . Anal . Behav . 6 , 299 300 -LRB- 1963 -RRB- . 36 . Davenport , J. W. , Goodrich , K. P. & Hagguist , W. W. Effects of magnitude of reinforcement in Macaca speciosa . Psych . Sci . 4 , 187 188 -LRB- 1966 -RRB- . 37 . Bechara , A. , Damasio , H. , Tranel , D. & Anderson , S. Dissociation of working memory from decision making within human prefrontal cortex . J. Neurosci . 18 , 428 437 -LRB- 1998 -RRB- . Acknowledgements . We thank J. Mones and S. Corathers for technical assistance ; W. T. Newsome , E. M. Brannon , M. Gazzaniga , A. Handel and J. A. Movshon for comments on the manuscript and experiments ; and M. N. Shadlen and W. T. Newsome for advice on analysis . This project was funded by the NEI -LRB- M.L.P. -RRB- and the McKnight Foundation -LRB- P.W.G. -RRB- . Correspondence and requests for materials should be addressed to M.L.P. -LRB- e-mail : mplatt@cns.nyu.edu -RRB- . Methods Single intraparietal neurons were studied in the brains of three rhesus monkeys trained to perform eye movements , in response to visual cues , for a fruit juice reward . Standard electrophysiological , behavioural and histological techniques were employed , as described34 . All procedures were approved by the New York University Animal Care and Use Committee and were in compliance with the Public Health Service 's Guide for the Care and Use of Animals . Each experiment began with the electrophysiological isolation of a single intraparietal neuron . Animals were then instructed to make a series of 50 200 eye movements from a central starting position to successively illuminated LEDs randomly selected from among several hundred LEDs located at different peripheral positions . We used these movements to identify an eccentric LED that elicited a movement for which the neuron was maximally active -LRB- a movement into the response eld -RRB- and an eccentric LED that elicited a movement for which the neuron was minimally active -LRB- a movement out of the response eld -RRB- . Experiment 1 . For experiment 1 , this mapping procedure was followed by a series of cued saccade trials in total darkness . Each of these trials began with the illumination of a yellow LED located directly in front of the animal . A random interval -LRB- 200 500 ms -RRB- after the subject had aligned his gaze -LRB- 62 8 -RRB- with this LED , the two eccentric yellow response LEDs identi ed by our mapping procedure were illuminated . After another random interval -LRB- 200 800 ms -RRB- , the central LED changed colour to either red or green , instructing the subject that a saccade aligning gaze with either the upper or lower eccentric LED , respectively , would be rewarded with a drop of juice . After a nal random interval -LRB- 200 800 ms -RRB- , the central LED was extinguished and the monkey was rewarded if he shifted gaze into alignment with -LRB- 64 8 -RRB- the correct eccentric LED . We then determined whether the activation of intraparietal neurons was modulated by systematic changes in the gain expected from each possible movement or the probability that each of the two possible movements would be instructed . Animals typically performed this task correctly on about 90 % of trials . Expected gain . While each of 40 neurons was studied we varied , in 3 7 sequential blocks of about 100 trials each , the amount of juice reward the animals received both for movements into and movements out of the response eld -LRB- expected gain -RRB- . Across blocks , the sum of the juice available from these two possible movements was held constant , and the probability that either response would be instructed on a given trial -LRB- outcome probability -RRB- was held constant at 0.5 . An experiment typically began with a block in which each movement was reinforced equally , and then pairs of blocks of complementary high gain/low gain conditions were randomly interleaved . Outcome probability . While each of 20 neurons was studied we varied , in 5 7 sequential blocks of about 100 trials each , the probability that each of the two responses would be instructed while the gain expected from each movement was held constant . A block in which each movement was instructed equally often was typically run rst , and then pairs of blocks with complementary high probability/low probability conditions were randomly interleaved . Experiment 2 . Thirty-six neurons were studied in two rhesus monkeys performing saccadic choice trials , which were similar to cued saccade trials but had no centrally located colour cue . Each of these trials began with the 238 1999 Macmillan Magazines Ltd NATURE | VOL 400 | 15 JULY 1999 | www.nature.com

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O B I O O O O B O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

neurophysiology
decisionmaking
decision making
reward
lip
monkey