{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T04:14:25Z","timestamp":1781583265999,"version":"3.54.5"},"reference-count":59,"publisher":"MIT Press - Journals","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Neural Computation"],"published-print":{"date-parts":[[2010,2]]},"abstract":"<jats:p> Population rate or activity equations are the foundation of a common approach to modeling for neural networks. These equations provide mean field dynamics for the firing rate or activity of neurons within a network given some connectivity. The shortcoming of these equations is that they take into account only the average firing rate, while leaving out higher-order statistics like correlations between firing. A stochastic theory of neural networks that includes statistics at all orders was recently formulated. We describe how this theory yields a systematic extension to population rate equations by introducing equations for correlations and appropriate coupling terms. Each level of the approximation yields closed equations; they depend only on the mean and specific correlations of interest, without an ad hoc criterion for doing so. We show in an example of an all-to-all connected network how our system of generalized activity equations captures phenomena missed by the mean field rate equations alone. <\/jats:p>","DOI":"10.1162\/neco.2009.02-09-960","type":"journal-article","created":{"date-parts":[[2009,10,23]],"date-time":"2009-10-23T15:09:34Z","timestamp":1256310574000},"page":"377-426","source":"Crossref","is-referenced-by-count":131,"title":["Systematic Fluctuation Expansion for Neural Network Activity Equations"],"prefix":"10.1162","volume":"22","author":[{"given":"Michael A.","family":"Buice","sequence":"first","affiliation":[{"name":"Laboratory of Biological Modeling\/NIDDK\/NIH, Bethesda, MD 20892, U.S.A."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jack D.","family":"Cowan","sequence":"additional","affiliation":[{"name":"University of Chicago, Departments of Mathematics and Neurology, Chicago, IL 60637, U.S.A."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Carson C.","family":"Chow","sequence":"additional","affiliation":[{"name":"Laboratory of Biological Modeling\/NIDDK\/NIH, Bethesda, MD 20892, U.S.A."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"281","reference":[{"key":"B1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.48.1483"},{"key":"B2","doi-asserted-by":"publisher","DOI":"10.1007\/BF00339367"},{"key":"B3","doi-asserted-by":"publisher","DOI":"10.1007\/BF00337259"},{"key":"B4","doi-asserted-by":"publisher","DOI":"10.1093\/cercor\/7.3.237"},{"key":"B5","doi-asserted-by":"publisher","DOI":"10.1088\/0954-898X\/8\/4\/003"},{"key":"B6","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.32.1007"},{"key":"B7","doi-asserted-by":"publisher","DOI":"10.1098\/rsta.2007.2092"},{"key":"B8","doi-asserted-by":"publisher","DOI":"10.1523\/JNEUROSCI.23-35-11167.2003"},{"key":"B9","doi-asserted-by":"publisher","DOI":"10.1523\/JNEUROSCI.1199-07.2007"},{"key":"B10","doi-asserted-by":"publisher","DOI":"10.1098\/rstb.2000.0769"},{"key":"B11","doi-asserted-by":"publisher","DOI":"10.1023\/A:1008925309027"},{"key":"B12","doi-asserted-by":"publisher","DOI":"10.1162\/089976699300016179"},{"key":"B13","doi-asserted-by":"publisher","DOI":"10.1162\/neco.2006.18.5.1066"},{"key":"B14","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.76.031118"},{"key":"B15","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.75.051919"},{"key":"B16","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.0401906101"},{"key":"B17","doi-asserted-by":"publisher","DOI":"10.1109\/TSMC.1983.6313075"},{"key":"B18","doi-asserted-by":"publisher","DOI":"10.1007\/s00422-005-0574-y"},{"key":"B19","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevD.10.2428"},{"key":"B20","volume-title":"Advances in neural information processing systems","volume":"3","author":"Cowan J. D.","year":"1991"},{"key":"B21","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/9\/9\/008"},{"key":"B22","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/9\/9\/009"},{"key":"B23","doi-asserted-by":"publisher","DOI":"10.1162\/neco.2009.02-08-710"},{"key":"B24","doi-asserted-by":"publisher","DOI":"10.1088\/0034-4885\/61\/4\/002"},{"key":"B25","doi-asserted-by":"publisher","DOI":"10.1007\/BF00336965"},{"key":"B26","doi-asserted-by":"publisher","DOI":"10.1162\/089976602320264015"},{"key":"B27","doi-asserted-by":"publisher","DOI":"10.1162\/089976699300016601"},{"key":"B28","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.51.738"},{"key":"B29","doi-asserted-by":"publisher","DOI":"10.1162\/089976600300015899"},{"key":"B30","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511815706"},{"key":"B31","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.50.3171"},{"key":"B32","doi-asserted-by":"publisher","DOI":"10.1162\/089976600300015529"},{"key":"B33","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.86.5.1698"},{"key":"B34","first-page":"1","volume":"98","author":"Hildebrand E. J.","year":"2006","journal-title":"Physical Review Letters"},{"key":"B35","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.79.8.2554"},{"key":"B36","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.81.10.3088"},{"key":"B38","doi-asserted-by":"publisher","DOI":"10.1016\/j.aop.2004.09.011"},{"key":"B39","doi-asserted-by":"publisher","DOI":"10.1162\/089976601750264974"},{"key":"B40","doi-asserted-by":"publisher","DOI":"10.1088\/1742-5468\/2007\/07\/P07024"},{"key":"B41","doi-asserted-by":"publisher","DOI":"10.1162\/neco.2007.19.8.2032"},{"key":"B42","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.8.423"},{"key":"B43","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.66.051917"},{"key":"B44","volume-title":"Introduction to plasma theory","author":"Nicholson D. R.","year":"1992"},{"key":"B45","doi-asserted-by":"publisher","DOI":"10.1023\/A:1008912914816"},{"key":"B46","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.48.2259"},{"key":"B47","doi-asserted-by":"publisher","DOI":"10.1051\/jphys:019850046090146900"},{"key":"B48","doi-asserted-by":"publisher","DOI":"10.1007\/BF00317939"},{"key":"B49","doi-asserted-by":"publisher","DOI":"10.1162\/089976600300015835"},{"key":"B50","doi-asserted-by":"publisher","DOI":"10.1162\/089976602320264024"},{"key":"B51","doi-asserted-by":"publisher","DOI":"10.1523\/JNEUROSCI.13-01-00334.1993"},{"key":"B52","doi-asserted-by":"publisher","DOI":"10.1162\/089976606774841567"},{"key":"B53","doi-asserted-by":"publisher","DOI":"10.1162\/neco.2007.19.12.3262"},{"key":"B54","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/38\/17\/R01"},{"key":"B55","doi-asserted-by":"publisher","DOI":"10.1088\/0954-898X\/4\/3\/002"},{"key":"B56","doi-asserted-by":"publisher","DOI":"10.1126\/science.274.5293.1724"},{"key":"B57","doi-asserted-by":"publisher","DOI":"10.1162\/089976698300017214"},{"key":"B58","doi-asserted-by":"publisher","DOI":"10.1016\/S0006-3495(72)86068-5"},{"key":"B59","doi-asserted-by":"publisher","DOI":"10.1007\/BF00288786"},{"key":"B60","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780198509233.001.0001"}],"container-title":["Neural Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mitpressjournals.org\/doi\/pdf\/10.1162\/neco.2009.02-09-960","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,3,12]],"date-time":"2021-03-12T21:37:16Z","timestamp":1615585036000},"score":1,"resource":{"primary":{"URL":"https:\/\/direct.mit.edu\/neco\/article\/22\/2\/377-426\/7540"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,2]]},"references-count":59,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2010,2]]}},"alternative-id":["10.1162\/neco.2009.02-09-960"],"URL":"https:\/\/doi.org\/10.1162\/neco.2009.02-09-960","relation":{},"ISSN":["0899-7667","1530-888X"],"issn-type":[{"value":"0899-7667","type":"print"},{"value":"1530-888X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,2]]}}}