[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81554-en":3,"doc-seo-81554-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},81554,549758146520,"Patrick","https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470",8,"Research & Report","Statistics of correlations in nonlinear recurrent neural networks","Statistics of correlations serve as key descriptors of collective dynamics in recurrent neural networks. The work derives exact correlation-statistics expressions for nonlinear recurrent networks in the large-N neuron limit, including systematic 1/N corrections, under Gaussian quenched disorder. A path-integral formulation reduces stochastic dynamics to a small set of collective variables for efficient computation. Nonlinear activation interactions resolve linear-theory instabilities, producing a strictly positive participation dimension. Analytical results are provided for power-law and Padé-based activations, validated by numerical simulations and contrasted with annealed-disorder studies to motivate a conjecture for colored noise.","arXiv :2510 .21742v3 [ q-bio .NC] 10 Jul 2026  \nStatistics of correlations in nonlinear recurrent  \nneural networks  \nGermán Mato 1 ,2 , Facundo Rigatuso 1 ,2 , Gonzalo Torroba 1 ,2  \n1 Instituto Balseiro, UNCuyo and CNEA  \n2 Centro Atómico Bariloche and CONICET  \nS. C. de Bariloche, Río Negro, R8402AGP, Argentina  \nAbstract  \nThe statistics of correlations are central quantities characterizing the collective dynamics of recurrent neural networks. We derive exact expressions for the statistics of correlations of nonlinear recurrent networks in the limit of a large number N of neurons, including systematic 1/N corrections, in the regime of Gaussian quenched disorder. Our approach uses a path-integral representation of the network’s stochastic dynamics, which reduces the description to a few collective variables and enables efficient computation. This generalizes previous results on linear networks to include a wide family of nonlinear activation functions, which enter as interaction terms in the path integral. These interactions can resolve the instability of the linear theory and yield a strictly positive participation dimension. We present explicit results for power-law activations, revealing scaling behavior controlled by the network coupling. In addition, we introduce a class of activation functions based on Padé approximants and provide analytic predictions for their correlation statistics. Numerical simulations confirm our theoretical results with excellent agreement. We also compare with previous works that have studied the complementary case with annealed disorder, and based on this we propose a conjectural self-consistent equation for the more general case of colored noise.  \nKeywords: neural population dynamics; recurrent neural network; nonlinear dynamics  \nContents  \n1 Introduction 1  \n2 Model and path integral representation 4  \n2.1 Equilibrium partition function representation .................. 5  \n2.2 Collective fields .................................. 7  \n3 Large N solution 8  \n3.1 1/N and source expansion ............................ 9  \n3.2 Effects of nonlinearities .............................. 10  \n3.3 Saddle point description of neural correlation functions ............ 12  \n3.4 Dimension of participation ............................ 16  \n4 Applications 17  \n4.1 Power-law activations ............................... 18  \n4.2 Padé activations and numerics .......................... 20  \n4.2.1 Activation function with p = 0 ...................... 21  \n4.2.2 Activation function with p = 1/2 .................... 23  \n4.3 Comparison to other works and annealed vs quenched disorder ........ 24  \n5 Conclusions 26  \nA Correlations of neural inputs 30  \nB Network simulations 33  \nC Derivation of the effective action at large N at order J 2 34  \n1 Introduction  \nCorrelations of neural activity are one of the main tools we have to study the structure and function of the nervous system. They provide crucial information by measuring the statistical interdependencies between the activity of different neurons or neural regions over time [1] . The interpretation of the correlations is not easy because they are not only controlled by direct interactions between the neurons but also by the global dynamical state of the system. For instance, theoretical analysis reveals that if the network is in an asynchronous state the cross-correlations are smaller than the autocorrelations by a factor 1/N , N being the network size [2] . This regime has been found to be present in a wide variety of systems [3, 4, 5, 6] under quite general conditions.  \nSmall cross-correlations are indeed found in cortical recordings both in spontaneous and active states [7, 8] . Even if the cross-correlations are small it has been recognized that they  \ncan be important for the coding and information transmission properties of networks [6] . For instance, in a study where correlated firing in MT (Middle Temporal area) was measured, it was found that spi","cbCaieCtRxqHFSo1","https://ap.wps.com/l/cbCaieCtRxqHFSo1","pdf",1575896,3,1,40,"English","en",105,"# Introduction\n# Model and path integral representation\n## Equilibrium partition function representation\n## Collective fields\n# Large N solution\n## 1/N and source expansion\n## Effects of nonlinearities\n## Saddle point description of neural correlation functions\n## Dimension of participation\n# Applications\n## Power-law activations\n## Padé activations and numerics\n## Comparison to other works and annealed vs quenched disorder\n# Conclusions\n# Correlations of neural inputs\n# Network simulations","[{\"question\":\"What correlations are analyzed, and why are they important for recurrent neural networks?\",\"answer\":\"The document focuses on correlation statistics of neural activity in recurrent networks. Correlations reveal statistical interdependencies across neurons over time and strongly affect collective dynamics and coding-relevant properties.\"},{\"question\":\"How does the paper compute correlation statistics in the large-N limit?\",\"answer\":\"It uses a path-integral representation of the network’s stochastic dynamics. The description is reduced to a few collective variables, and the approach yields exact expressions including systematic 1/N corrections under Gaussian quenched disorder.\"},{\"question\":\"What role do nonlinear activation functions play compared with linear theory?\",\"answer\":\"Nonlinear activation interactions appear in the path integral and can resolve instabilities present in the linear theory. They also lead to a strictly positive participation dimension, and explicit scaling results are given for power-law and Padé-based activations.\"}]",1784174285,101,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"statistics-of-correlations-in-nonlinear-recurrent-neural-networks","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/statistics-of-correlations-in-nonlinear-recurrent-neural-networks/81554/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What correlations are analyzed, and why are they important for recurrent neural networks?","Question",{"text":75,"@type":76},"The document focuses on correlation statistics of neural activity in recurrent networks. Correlations reveal statistical interdependencies across neurons over time and strongly affect collective dynamics and coding-relevant properties.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper compute correlation statistics in the large-N limit?",{"text":80,"@type":76},"It uses a path-integral representation of the network’s stochastic dynamics. The description is reduced to a few collective variables, and the approach yields exact expressions including systematic 1/N corrections under Gaussian quenched disorder.",{"name":82,"@type":73,"acceptedAnswer":83},"What role do nonlinear activation functions play compared with linear theory?",{"text":84,"@type":76},"Nonlinear activation interactions appear in the path integral and can resolve instabilities present in the linear theory. They also lead to a strictly positive participation dimension, and explicit scaling results are given for power-law and Padé-based activations.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,119,122,127,130,134],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":22,"slug":118},7,"Healthcare","healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]