{"id":34140,"date":"2022-05-03T22:26:42","date_gmt":"2022-05-03T14:26:42","guid":{"rendered":"https:\/\/swarma.org\/?p=34140"},"modified":"2022-05-03T22:26:42","modified_gmt":"2022-05-03T14:26:42","slug":"prl%e9%80%9f%e9%80%92%ef%bc%9a%e9%87%8d%e6%95%b4%e5%8c%96%e7%90%86%e8%ae%ba%e6%8f%8f%e8%bf%b0%e7%a5%9e%e7%bb%8f%e7%bd%91%e7%bb%9c%e5%8a%a8%e5%8a%9b%e5%ad%a6","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=34140","title":{"rendered":"PRL\u901f\u9012\uff1a\u91cd\u6574\u5316\u7406\u8bba\u63cf\u8ff0\u795e\u7ecf\u7f51\u7edc\u52a8\u529b\u5b66"},"content":{"rendered":"
\n
\n
\n
\n
\n

<\/p>\n<\/section>\n

\n
\n
\n
\n
\n
\n

<\/section>\n<\/section>\n
\n
\n

\u6458\u8981<\/strong><\/p>\n<\/section>\n<\/section>\n

\n

<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n
\n
\n
\n

\u4e34\u754c\u6027\u4e0e\u6700\u4f73\u8ba1\u7b97\u80fd\u529b\u6709\u7740\u6df1\u523b\u7684\u8054\u7cfb\u3002\u7136\u800c\uff0c\u5230\u76ee\u524d\u4e3a\u6b62\uff0c\u7531\u4e8e\u7f3a\u4e4f\u5927\u8111\u4e34\u754c\u52a8\u529b\u5b66\u7684\u91cd\u6574\u5316\u7406\u8bba\uff0c\u5bf9\u8fd9\u79cd\u5f62\u5f0f\u7684\u751f\u7269\u4fe1\u606f\u5904\u7406\u7684\u7406\u89e3\u4ec5\u9650\u4e8e\u5e73\u5747\u573a\u7ed3\u679c\u3002\u8fd9\u4e9b\u65b9\u6cd5\u5ffd\u7565\u4e86\u4e34\u754c\u7cfb\u7edf\u7684\u4e00\u4e2a\u5173\u952e\u7279\u5f81\uff1a\u8de8\u8d8a\u6240\u6709\u957f\u5ea6\u5c3a\u5ea6\u7684\u81ea\u7531\u5ea6\u4e4b\u95f4\u5b58\u5728\u76f8\u4e92\u4f5c\u7528\uff0c\u8fd9\u662f\u590d\u6742\u975e\u7ebf\u6027\u8ba1\u7b97\u6240\u9700\u8981\u7684\u3002\u6211\u4eec\u63d0\u51fa\u4e86\u4e00\u79cd\u795e\u7ecf\u573a\u7406\u8bba\u7684\u91cd\u6574\u5316\u7406\u8bba\uff0c\u5373\u968f\u673a Wilson-Cowan \u6a21\u578b\u3002\u6211\u4eec\u8ba1\u7b97\u4e86\u8026\u5408\u6d41\uff0c\u5b83\u5728\u4e0d\u65ad\u589e\u52a0\u7684\u957f\u5ea6\u5c3a\u5ea6\u4e0a\u53c2\u6570\u5316\u76f8\u4e92\u4f5c\u7528\u3002\u5c3d\u7ba1\u4e0e Kardar-Parisi-Zhang \u6a21\u578b\u6709\u76f8\u4f3c\u4e4b\u5904\uff0c\u4f46\u8be5\u7406\u8bba\u5c5e\u4e8e Gell-Mann-Low \u7c7b\u578b\uff0c\u662f\u53ef\u91cd\u6574\u5316\u91cf\u5b50\u573a\u8bba\u7684\u5178\u578b\u5f62\u5f0f\u3002\u5728\u8fd9\u91cc\uff0c\u975e\u7ebf\u6027\u8026\u5408\u6d88\u5931\uff0c\u6d41\u5411\u9ad8\u65af\u56fa\u5b9a\u70b9\uff0c\u4f46\u5448\u5bf9\u6570\u6162\u5316\uff0c\u56e0\u6b64\u5728\u5927\u591a\u6570\u5c3a\u5ea6\u4e0a\u4ecd\u7136\u6709\u6548\u3002\u6211\u4eec\u5c55\u793a\u4e86\u8fd9\u79cd\u4e34\u754c\u7684\u76f8\u4e92\u4f5c\u7528\u7ed3\u6784\uff0c\u4ee5\u5b9e\u73b0\u4fe1\u606f\u5b58\u50a8\u7684\u6700\u4f73\u7ebf\u6027\u548c\u8ba1\u7b97\u6240\u9700\u7684\u975e\u7ebf\u6027\u4e4b\u95f4\u7684\u7406\u60f3\u6743\u8861\u3002<\/span><\/strong><\/span><\/p>\n

 <\/p>\n

\u7814\u7a76\u9886<\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong>\u57df\uff1a<\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/span>\u795e\u7ecf\u7f51\u7edc\u52a8\u529b\u5b66\uff0c\u91cd\u6574\u5316\uff0c\u91cf\u5b50\u573a\u8bba<\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

\n
\n
\n
<\/svg><\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n
\n
<\/strong><\/section>\n<\/section>\n
\n

\u6f58\u4f73\u680b<\/span> <\/span>| \u4f5c\u8005<\/strong><\/span><\/p>\n

\u6881\u91d1<\/span> <\/span>| <\/strong><\/span>\u5ba1\u6821<\/strong><\/span><\/p>\n

\u9093\u4e00\u96ea<\/span> | \u7f16\u8f91<\/strong><\/span><\/strong><\/span><\/p>\n


<\/p>\n


<\/p>\n<\/section>\n

<\/span><\/p>\n

\n
\n
\n
\u8bba\u6587\u9898\u76ee\uff1a<\/span><\/section>\n
Gell-Mann\u2013Low Criticality in Neural Networks<\/span><\/section>\n
\u8bba\u6587\u94fe\u63a5\uff1a<\/span><\/section>\n
https:\/\/journals.aps.org\/prl\/abstract\/10.1103\/PhysRevLett.128.168301<\/span><\/span><\/section>\n<\/section>\n<\/section>\n<\/blockquote>\n


<\/p>\n

<\/h1>\n

\n
\n
\n
\n
\n

<\/section>\n

<\/section>\n
\n
<\/svg><\/section>\n<\/section>\n<\/section>\n
\n

1. \u4e34\u754c\u7684\u5927\u8111<\/span><\/strong><\/strong><\/p>\n<\/section>\n

\n

<\/section>\n

<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


<\/p>\n

\n
\n

\u4e34\u754c\u70b9\u548c\u4fe1\u606f\u5904\u7406\u6709\u5f88\u6df1\u7684\u5173\u7cfb\u3002\u4f8b\u5982\uff0c\u5bf9\u6700\u96be\u89e3\u51b3\u7684\u7ec4\u5408\u4f18\u5316\u95ee\u9898\u7684\u7edf\u8ba1\u63cf\u8ff0\uff0c\u5c31\u5904\u4e8e\u76f8\u53d8\u7684\u4e34\u754c\u70b9[1,2]\u3002\u5927\u8111\u6d3b\u52a8\u4e5f\u663e\u793a\u51fa\u4e34\u754c\u6027\uff0c\u8fd9\u4f3c\u4e4e\u53ef\u4ee5\u4f18\u5316\u7f51\u7edc\u7684\u8ba1\u7b97\u5c5e\u6027\u3002\u4e3a\u4e86\u6709\u6548\u5730\u5904\u7406\u4fe1\u606f\uff0c\u5927\u8111\u88ab\u8ba4\u4e3a\u662f\u5728\u4e24\u76f8\u4e4b\u95f4\u7684\u4e34\u754c\u70b9\u4e0a\u8fd0\u884c\u3002\u5728\u4e34\u754c\u70b9\u4e0a\uff0c\u795e\u7ecf\u5143\u7f51\u7edc\u8db3\u591f\u7a33\u5b9a\uff0c\u80fd\u591f\u53ef\u9760\u5730\u5b58\u50a8\u4fe1\u606f\uff0c\u4f46\u53c8\u8db3\u591f\u654f\u611f\uff0c\u80fd\u591f\u8fc5\u901f\u5411\u5927\u8111\u7684\u5176\u4ed6\u9065\u8fdc\u90e8\u4f4d\u53d1\u9001\u4fe1\u53f7<\/strong>\u3002\u73b0\u5728\uff0c\u4f7f\u7528\u91cf\u5b50\u573a\u8bba\u4e2d\u7684\u65b9\u6cd5\uff0c\u5fb7\u56fd\u5c24\u5229\u5e0c\u7814\u7a76\u4e2d\u5fc3\u7684 Moritz Helias \u548c\u4ed6\u7684\u540c\u4e8b\u8bc1\u5b9e\uff0c\u8fd9\u4e9b\u4e34\u754c\u70b9\u5728\u5927\u8111\u52a8\u529b\u5b66\u7684\u7ecf\u5178\u6a21\u578b\u4e2d\u786e\u5b9e\u5b58\u5728\u3002<\/span><\/p>\n

 <\/span><\/p>\n

\u8be5\u56e2\u961f\u8003\u8651\u4e86 Wilson-Cowan \u6a21\u578b\uff0c\u5728\u8be5\u6a21\u578b\u4e2d\uff0c\u795e\u7ecf\u5143\u96c6\u5408\u88ab\u5916\u90e8\u523a\u6fc0\u6216\u4e0e\u90bb\u5c45\u7684\u76f8\u4e92\u4f5c\u7528\u6fc0\u53d1\u3002\u5728\u4ee5\u524d\u5bf9\u8fd9\u4e2a\u6a21\u578b\u7684\u7814\u7a76\u4e2d\uff0c\u7814\u7a76\u4eba\u5458\u4f7f\u7528\u7b80\u5316\u7684\u5e73\u5747\u573a\u8fd1\u4f3c\uff0c\u4f46\u662f\u5b83\u4e0d\u80fd\u89e3\u91ca\u5728\u5927\u8111\u4e2d\u89c2\u5bdf\u5230\u7684\u5168\u90e8\u590d\u6742\u73b0\u8c61\u3002\u4e8b\u5b9e\u4e0a\uff0c\u7814\u7a76\u4eba\u5458\u53d1\u73b0 Wilson-Cowan \u6a21\u578b\u786e\u5b9e\u53ef\u4ee5\u5b58\u5728\u4e34\u754c\u70b9\u3002\u5728\u4e8c\u7ef4\u7a7a\u95f4\uff0c\u5373\u5bf9\u5e94\u4e8e\u76ae\u8d28\u7f51\u7edc\u7684\u5e73\u9762\u7ec4\u7ec7\u4e2d\uff0c\u5e73\u5747\u573a\u7406\u8bba\u5931\u6548\u3002<\/span><\/p>\n

 <\/span><\/p>\n

<\/p>\n

\u56fe1. (a) \u6a21\u578b\u539f\u7406\u56fe\u3002(b) \u65b9\u5dee\u3002\u84dd\u8272\u83f1\u5f62\u4e3a\u6a21\u62df\u60c5\u51b5\uff1b\u6df1\u84dd\u8272\u865a\u7ebf\u4e3a\u7ebf\u6027\u7cfb\u7edf\uff1b\u7070\u8272\u5b9e\u7ebf\u4e3a\u91cd\u6574\u5316\u7406\u8bba\u7ed9\u51fa\u3002(c) \u8026\u5408\u6d41\u3002\u8fc7\u6e21\u7ebf\uff08\u9ed1\u8272\u865a\u7ebf\uff09\u5c06\u6536\u655b\u533a\uff08\u7eff\u8272\uff09\u548c\u53d1\u6563\u533a\uff08\u7ea2\u8272\uff09\u5206\u5f00\u3002\u975e\u7269\u7406\u7a33\u6001\u7684\u533a\u57df\u4e3a\u6df1\u7ea2\u8272\u3002(d) \u8bb0\u5fc6\u3002\u5bf9\u4e8e\u4e0d\u540c\u5f3a\u5ea6\u975e\u7ebf\u6027\u9ad8\u65af\u8f93\u5165\u7684\u91cd\u5efa\u7cbe\u5ea6\uff080\uff1a\u5b8c\u5168\u9057\u5fd8\uff1b1\uff1a\u5b8c\u7f8e\u91cd\u5efa\uff09\u968f\u65f6\u95f4\u7684\u53d8\u5316\u3002(e) \u5206\u7c7b\u3002\u6b63\u786e\u5206\u914d\u7ed93\u4f4d\u5b57\u7b26\u4e32\u7684\u5947\u5076\u6027\uff08\u84dd\u8272\u66f2\u7ebf\uff09\u548c\u91cd\u590d\u8bad\u7ec3\u7684\u6807\u51c6\u504f\u5dee\uff08\u9634\u5f71\u533a\u57df\uff09\u3002\u8bfb\u51fa\u65f6\u6700\u6162\uff08\u7ea2\u8272\uff09\u3001\u4e2d\u95f4\uff08\u6a59\u8272\uff09\u548c\u6700\u5feb\uff08\u7eff\u8272\uff09\u6a21\u5f0f\u7684\u8870\u51cf\u65f6\u95f4\u5c3a\u5ea6\u3002<\/span><\/section>\n

 <\/span><\/p>\n


<\/p>\n

\n
\n
\n
\n
\n

<\/section>\n

<\/section>\n
\n
<\/svg><\/section>\n<\/section>\n<\/section>\n
\n

2. \u8d85\u8d8a\u5e73\u5747\u573a\uff0c<\/span><\/strong><\/strong><\/p>\n

\u91cd\u6574\u5316\u7406\u8bba\u63cf\u8ff0\u795e\u7ecf\u7f51\u7edc\u52a8\u529b\u5b66<\/span><\/strong><\/strong><\/p>\n<\/section>\n

\n

<\/section>\n

<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


<\/p>\n

Moritz Helias \u548c\u4ed6\u7684\u56e2\u961f\u901a\u8fc7\u5e94\u7528\u4e00\u79cd\u53eb\u505a\u91cd\u6574\u5316\u7684\u6280\u672f\u8d85\u8d8a\u4e86\u8fd9\u79cd\u8fd1\u4f3c\uff0c\u8fd9\u79cd\u6280\u672f\u5728\u91cf\u5b50\u573a\u8bba\u4e2d\u901a\u5e38\u7528\u4e8e\u5904\u7406\u5947\u70b9<\/span>\uff08singularity\uff09<\/span>\u2014\u2014\u5177\u6709\u65e0\u7a77\u503c\u7684\u91cf\u3002\u8be5\u56e2\u961f\u7684\u6a21\u578b\u8fd8\u6d89\u53ca\u795e\u7ecf\u5143\u4e4b\u95f4\u76f8\u4e92\u4f5c\u7528\u5f3a\u5ea6\u7684\u533a\u522b\u3002\u4f46\u7814\u7a76\u4eba\u5458\u53d1\u73b0\uff0c\u5728\u91cd\u6574\u5316\u4e4b\u540e\uff0c\u8fd1\u5904\u548c\u8fdc\u5904\u7684\u795e\u7ecf\u5143\u90fd\u80fd\u6709\u6548\u5730\u76f8\u4e92\u6c9f\u901a\uff0c\u540c\u65f6\u4ecd\u7136\u4fdd\u7559\u4e86\u5b58\u50a8\u8bb0\u5fc6\u7684\u80fd\u529b\u3002\u8fd9\u79cd\u5728\u5c0f\u8ddd\u79bb\u548c\u5927\u8ddd\u79bb\u5c3a\u5ea6\u4e0a\u7684\u534f\u8c03\u662f\u4e34\u754c\u6027\u7684\u4e00\u4e2a\u6807\u5fd7\u3002<\/strong><\/span><\/p>\n

 <\/span><\/p>\n

\u7814\u7a76\u4eba\u5458\u53d1\u73b0\u5e76\u8bc1\u660e\u4e86 Gell-Mann-Low \u7c7b\u578b\u7684\u4e34\u754c\u6027\u3002\u8fd9\u79cd\u4e34\u754c\u6027\u5bf9\u5927\u8111\u8ba1\u7b97\u6781\u4e3a\u5173\u952e\u3002\u4fe1\u606f\u5904\u7406\u53ef\u80fd\u53d7\u76ca\u4e8e\u7ebf\u6027\u884c\u4e3a\u4e0e\u975e\u7ebf\u6027\u884c\u4e3a\u4e4b\u95f4\u7684\u5e73\u8861<\/strong>\uff0c\u524d\u8005\u652f\u6301\u4fe1\u606f\u5b58\u50a8\u548c\u4f20\u9012\uff0c\u540e\u8005\u5219\u662f\u8ba1\u7b97\u6240\u5fc5\u9700\u7684\u3002Gell-Mann-Low \u4e34\u754c\u6027\u901a\u8fc7\u4f7f\u4e34\u754c\u70b9\u4f4d\u4e8e\u5e73\u5747\u573a\u548c\u5f3a\u8026\u5408\u56fa\u5b9a\u70b9\u4e4b\u95f4\uff0c\u5b9e\u73b0\u4e86\u8fd9\u79cd\u5e73\u8861\u3002<\/span><\/p>\n

 <\/span><\/p>\n

\u4ed6\u4eec\u9996\u5148\u5173\u6ce8\u7684\u662f\u8bb0\u5fc6\u3002\u4e3a\u4e86\u5177\u8c61\u5316\uff0c\u8003\u8651\u8d39\u820d\u5c14\u8bb0\u5fc6\u66f2\u7ebf\u3002\u5728\u7ebf\u6027\u60c5\u51b5\u4e0b\uff0c\u5b83\u968f\u65f6\u95f4\u8870\u51cf\u4e3at-2<\/sup><\/span>\uff0c\u8fd9\u662f\u6700\u4f18\u7684[3]\u3002\u76f4\u89c2\u5730\u8bf4\uff0c\u975e\u7ebf\u6027\u884c\u4e3a\u5f15\u8d77\u7684\u4fe1\u53f7\u5931\u771f\u4f7f\u5f97\u68c0\u7d22\u8fc7\u53bb\u8f93\u5165\u7684\u4fe1\u606f\u66f4\u52a0\u56f0\u96be<\/span>\uff08\u5f53\u7136\uff0c\u4e00\u4e9b\u7f51\u7edc\u7ed3\u6784\u53ef\u80fd\u88ab\u5fae\u8c03\u4e3a\u4e0e\u975e\u7ebf\u6027\u7684\u6700\u4f73\u76f8\u4e92\u4f5c\u7528\uff0c\u4ece\u800c\u589e\u52a0\u8bb0\u5fc6\uff0c\u4f8b\u5982\u901a\u8fc7\u4e3b\u52a8\u566a\u58f0\u6291\u5236[4]\uff0c\u4f46\u8fd9\u5728\u6a21\u578b\u4e2d\u6ca1\u6709\u53d1\u751f\uff09<\/span>\u3002\u5728 Gell-Mann-Low \u60c5\u51b5\u4e2d\uff0c\u975e\u7ebf\u6027\u6d41\u53d8\u4e3a\u96f6\uff0c\u800c\u4e0d\u662f\u53d8\u6210\u4e00\u4e2a\u5f3a\u8026\u5408\u7684\u56fa\u5b9a\u70b9\u3002<\/span><\/p>\n

 <\/span><\/p>\n

\u7136\u800c\uff0c\u4e0e\u5e73\u5747\u573a\u60c5\u51b5\u76f8\u53cd\uff0c\u975e\u7ebf\u6027\u76f8\u4e92\u4f5c\u7528\u53ea\u4ee5\u5bf9\u6570\u901f\u5ea6\u6d88\u5931\u3002\u56e0\u6b64\uff0c\u5b83\u4eec\u5b9e\u9645\u4e0a\u5728\u6240\u6709\u957f\u5ea6\u5c3a\u5ea6\u4e0a\u90fd\u662f\u6709\u6548\u7684\uff0c\u5141\u8bb8\u7cfb\u7edf\u7684\u81ea\u7531\u5ea6\u76f8\u4e92\u4f5c\u7528\u5e76\u5171\u540c\u8fdb\u884c\u8ba1\u7b97\u3002\u4ed6\u4eec\u901a\u8fc7\u8bad\u7ec3\u4e24\u4e2a\u7ebf\u6027\u51fd\u6570\u6765\u6b63\u786e\u5206\u7c7b3\u4f4d\u5b57\u7b26\u4e32\u7684\u5947\u5076\u6027\uff0c\u4ece\u800c\u8bf4\u660e\u975e\u7ebf\u6027\u76f8\u4e92\u4f5c\u7528\u7684\u6027\u8d28[5]\u3002<\/span><\/p>\n

 <\/span><\/p>\n

\u7814\u7a76\u4eba\u5458\u63d0\u4f9b\u4e86\u4e00\u4e2a\u795e\u7ecf\u7f51\u7edc\u52a8\u529b\u5b66\u7684\u91cd\u6574\u5316\u7406\u8bba\uff0c\u4ee5\u63ed\u793a\u8de8\u5c3a\u5ea6\u7684\u975e\u7ebf\u6027\u76f8\u4e92\u4f5c\u7528\u7684\u7ed3\u6784<\/strong>\uff0c\u8fd9\u662f\u8fc4\u4eca\u4e3a\u6b62\u5e73\u5747\u573a\u65b9\u6cd5\u6240\u4e0d\u80fd\u505a\u5230\u7684\u3002\u8fd9\u4e2a\u6846\u67b6\u4e3a\u8d85\u8d8a\u5e73\u5747\u573a\u7684\u666e\u9002\u6027[6,7]\u6253\u5f00\u4e86\u5927\u95e8\uff0c\u8fd9\u4e9b\u666e\u9002\u6027\u7c7b\u53ef\u80fd\u9690\u85cf\u5728\u4e34\u754c\u5927\u8111\u6d3b\u52a8\u80cc\u540e\u3002\u4ed6\u4eec\u8ba4\u4e3a\uff0c\u8d85\u8d8a\u5e73\u5747\u573a\u7684\u884c\u4e3a\u662f\u8ba1\u7b97\u7684\u57fa\u7840\uff0c\u5e76\u63d0\u4f9b\u5de5\u5177\u6765\u5b9a\u91cf\u5904\u7406\u5904\u4e8e\u4e34\u754c\u70b9\u7684\u975e\u7ebf\u6027\u4fe1\u53f7\u8f6c\u6362\u3002\u5e94\u7528\u968f\u673a Wilson-Cowan \u6a21\u578b\uff0c\u4ed6\u4eec\u53d1\u73b0\u4e86\u4e00\u79cd\u65b0\u7684\u4e34\u754c\u70b9\u5f62\u5f0f\uff0c\u5b83\u5bf9\u751f\u7269\u5b66\u4e0a\u5408\u7406\u7684\u5916\u90e8\u9a71\u52a8\u662f\u9c81\u68d2\u7684\u3002\u5728\u4e8c\u7ef4\u60c5\u51b5\u4e0b\uff0c\u4e34\u754c\u884c\u4e3a\u662fGell-Mann-Low\u7c7b\u578b\u7684\uff1b\u4e0e\u5e73\u5747\u573a\u884c\u4e3a\u76f8\u53cd\uff0c\u975e\u7ebf\u6027\u8026\u5408\uff0c\u5373\u8ba1\u7b97\u7684\u6839\u672c\uff0c\u5728\u8fd9\u91cc\u4ecd\u7136\u4e0e\u6240\u6709\u957f\u5ea6\u5c3a\u5ea6\u76f8\u5173\u3002\u7136\u800c\uff0c\u5b83\u4eec\u7f13\u6162\u6d88\u5931\u7684\u6d41\u5e76\u4e0d\u6539\u53d8\u5e73\u5747\u573a\u4e34\u754c\u6307\u6570\u7684\u9636\u3002\u8fd9\u53ef\u80fd\u89e3\u91ca\u4e86\u4e3a\u4ec0\u4e48\u4ee5\u524d\u7684\u5de5\u4f5c\u53ef\u80fd\u5ffd\u7565\u4e86\u8d85\u51fa\u5e73\u5747\u573a\u7684\u884c\u4e3a\u3002<\/span><\/p>\n


<\/p>\n


<\/p>\n

\n
\n
\n
\n
\n

<\/section>\n

<\/section>\n
\n
<\/svg><\/section>\n<\/section>\n<\/section>\n
\n

3. \u5c55\u671b<\/span><\/strong><\/strong><\/p>\n<\/section>\n

\n

<\/section>\n

<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


<\/p>\n

\u63a2\u8ba8\u5728\u66f4\u590d\u6742\u7684\u6a21\u578b\u4e2d\u662f\u5426\u4f1a\u51fa\u73b0\u8fd9\u79cd\u6216\u5176\u4ed6\u7c7b\u578b\u7684\u4e34\u754c\uff0c\u5c06\u662f\u5f88\u6709\u8da3\u7684\u3002\u4f8b\u5982\uff0c\u5728\u5177\u6709\u591a\u4e2a\u76ae\u8d28\u5c42\u7684\u6a21\u578b\u4e2d\uff0c\u6d8c\u73b0\u884c\u4e3a\u53ef\u80fd\u4f1a\u5f71\u54cd\u6cbf\u76ae\u8d28\u8868\u9762\u7684\u4e34\u754c\u6027\u3002\u9664\u4e86\u8fd9\u91cc\u8003\u8651\u7684\u7a81\u89e6\u5f3a\u5ea6\u5916\uff0c\u5c42\u6b21\u7f51\u7edc\u7684\u7ed3\u6784\u4e5f\u662f\u4e00\u4e2a\u53c2\u6570\uff0c\u5b83\u53ef\u4ee5\u5c06\u7cfb\u7edf\u8c03\u6574\u5230\u4e34\u754c\u72b6\u6001\uff0c\u5e76\u6709\u8da3\u5730\u5c06\u5176\u6269\u5c55\u5230\u53c2\u6570\u7a7a\u95f4\u7684\u5e7f\u6cdb\u533a\u57df[8]\u3002\u4e00\u822c\u6765\u8bf4\uff0c\u8fd9\u91cc\u63d0\u51fa\u7684\u65b9\u6cd5\u9002\u7528\u4e8e\u4efb\u4f55\u968f\u673a\u52a8\u529b\u5b66\u7cfb\u7edf\uff1b\u7279\u522b\u662f\u795e\u7ecf\u7f51\u7edc\u6a21\u578b\uff0c\u5176\u4e2d\u4e00\u4e9b\u6a21\u578b\u751a\u81f3\u88ab\u8868\u8ff0\u4e3a\u573a\u8bba\u3002<\/span><\/p>\n

 <\/span><\/p>\n

\u91cd\u6574\u5316\u6280\u672f\u4e5f\u6b63\u88ab\u5e94\u7528\u4e8e\u4eba\u5de5\u795e\u7ecf\u7f51\u7edc[9,10]\u3002\u5728\u50a8\u5c42\u8ba1\u7b97\u7684\u80cc\u666f\u4e0b\uff0c\u4ed6\u4eec\u8ba4\u4e3aGell-Mann-Low\u4e34\u754c\u6027\u652f\u6301\u7ebf\u6027\u548c\u975e\u7ebf\u6027\u52a8\u529b\u5b66\u4e4b\u95f4\u7684\u8ba1\u7b97\u4e0a\u7684\u6700\u4f73\u5e73\u8861\u3002\u4ed6\u4eec\u7684\u5206\u6790\u4e5f\u53ef\u4ee5\u6269\u5c55\u5230\u5177\u6709\u53ef\u8bad\u7ec3\u7684\u9690\u85cf\u6743\u91cd\u7684\u9012\u5f52\u7f51\u7edc\uff0c\u8fd9\u4e9b\u7f51\u7edc\u9002\u5408\u4e8e\u573a\u8bba\u7684\u8868\u8ff0[11-13]\uff0c\u8fdb\u4e00\u6b65\u7814\u7a76\u4e34\u754c\u6027\u548c\u8ba1\u7b97\u4e4b\u95f4\u7684\u5173\u7cfb\u3002\u603b\u7684\u6765\u8bf4\uff0c\u8fd9\u91cc\u7684\u795e\u7ecf\u573a\u7406\u8bba\u3001\u975e\u5e73\u8861\u7edf\u8ba1\u7269\u7406\u5b66\u7684\u6838\u5fc3Kardar-Parisi-Zhang\u6a21\u578b\u548c\u91cf\u5b50\u573a\u8bba\u4e4b\u95f4\u7684\u8054\u7cfb\uff0c\u4e3a\u4ece\u8fd9\u4e9b\u9886\u57df\u8f6c\u79fb\u4e13\u4e1a\u77e5\u8bc6\u63d0\u4f9b\u4e86\u65b9\u6cd5\uff0c\u8fd9\u4e9b\u65b9\u6cd5\u5728\u4e0a\u8ff0\u9886\u57df\u88ab\u5e7f\u6cdb\u4f7f\u7528\u3002<\/span><\/p>\n

 <\/span><\/p>\n

\u867d\u7136\u6709\u5f3a\u6709\u529b\u7684\u8bc1\u636e\u8868\u660e\u5927\u8111\u4e2d\u5b58\u5728\u4e34\u754c\u72b6\u6001\uff0c\u4f46\u8fd8\u6ca1\u6709\u4eba\u5b8c\u5168\u89e3\u91ca\u4e3a\u4ec0\u4e48\u6211\u4eec\u7684\u5927\u8111\u4f1a\u4ee5\u8fd9\u79cd\u65b9\u5f0f\u8fd0\u4f5c\u3002\u8fd9\u4e2a\u65b0\u7ed3\u679c\u662f\u671d\u8fd9\u4e2a\u65b9\u5411\u8fc8\u51fa\u7684\u4e00\u6b65\u3002\u8be5\u56e2\u961f\u4e0b\u4e00\u6b65\u8ba1\u5212\u5728\u795e\u7ecf\u5143\u7684\u8fde\u63a5\u65b9\u5f0f\u4e2d\u52a0\u5165\u4e00\u4e9b\u968f\u673a\u6027\uff0c\u5c31\u50cf\u6211\u4eec\u5927\u8111\u4e2d\u7684\u60c5\u51b5\u4e00\u6837\u3002<\/span><\/p>\n

 <\/span><\/p>\n

 <\/span><\/p>\n

\u53c2\u8003\u6587\u732e\uff1a <\/span><\/strong>    <\/span><\/p>\n

[1] P. Cheeseman, B. Kanefsky, and W. M. Taylor, in Proceed- ings of the 12th International Joint Conference on Artificial Intelligence\u2014Volume 1 (Morgan Kaufmann Publishers Inc., San Francisco, CA, 1991), IJCAI\u201991, pp. 331\u2013337,  ISBN 1558601600.<\/span><\/p>\n

[2] L. Saitta, A. Giordana, and A. Cornue \u0301jols, Phase Transitions in Machine Learning (Cambridge University Press, Cambridge, England, 2011).<\/span><\/p>\n

[3] S. Ganguli, D. Huh, and H. Sompolinsky, Proc. Natl. Acad. Sci. U.S.A. 105, 18970 (2008).<\/span><\/p>\n

[4] T. Toyoizumi, Neural Comput. 24, 2678 (2012).<\/span><\/p>\n

[5] See Supplemental Material at <\/span>http:\/\/link.aps.org\/supplemental<\/span>10.1103\/PhysRevLett.128.168301 for further details on theoretical calculations, numerical simulations, and computational tasks. <\/span><\/span><\/p>\n

[6] P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49, 435 (1977).<\/span><\/p>\n

[7] U. C. Taeuber, Critical Dynamics: A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior (Cambridge University Press, Cambridge, England, 2014), ISBN 9780521842235. <\/span><\/p>\n

[8] P. Moretti and M. A. Mun\u0303oz, Nat. Commun. 4, 2521 (2013). <\/span><\/p>\n

[9] P. Mehta and D. J. Schwab, arXiv:1410.3831.<\/span><\/p>\n

[10] D. A. Roberts, S. Yaida, and B. Hanin, The Principles of Deep Learning Theory: An Effective Theory Approach to Understanding Neural Networks (Cambridge University Press, Cambridge, England, 2022).<\/span><\/p>\n

[11] H. Sompolinsky, A. Crisanti, and H. J. Sommers, Phys. Rev. Lett. 61, 259 (1988).<\/span><\/p>\n

[12] J. Schuecker, S. Goedeke, and M. Helias, Phys. Rev. X 8, 041029 (2018).<\/span><\/p>\n

[13] K. Krishnamurthy, T. Can, and D. J. Schwab, Phys. Rev. X 12, 011011 (2022). <\/span><\/p>\n


<\/p>\n<\/section>\n<\/h3>\n


<\/p>\n

\n
\n
\n

\u8bba\u6587 Abstract<\/strong><\/span><\/p>\n<\/section>\n<\/section>\n<\/section>\n

Criticality is deeply related to optimal computational capacity. The lack of a renormalized theory of critical brain dynamics, however, so far limits insights into this form of biological information processing to mean-field results. These methods neglect a key feature of critical systems: the interaction between degrees of freedom across all length scales, required for complex nonlinear computation. We present a renormalized theory of a prototypical neural field theory, the stochastic Wilson-Cowan equation. We compute the flow of couplings, which parametrize interactions on increasing length scales. Despite similarities with the Kardar-Parisi-Zhang model, the theory is of a Gell-Mann\u2013Low type, the archetypal form of a renormalizable quantum field theory. Here, nonlinear couplings vanish, flowing towards the Gaussian fixed point, but logarithmically slowly, thus remaining effective on most scales. We show this critical structure of interactions to implement a desirable trade-off between linearity, optimal for information storage, and nonlinearity, required for computation.<\/span><\/span><\/span><\/section>\n

<\/section>\n
\n


<\/p>\n<\/section>\n

\n
\n
\n
\n

\u795e\u7ecf\u52a8\u529b\u5b66\u6a21\u578b\u8bfb\u4e66\u4f1a<\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

\n


<\/p>\n<\/section>\n

\u968f\u7740\u7535\u751f\u7406\u5b66\u3001\u7f51\u7edc\u5efa\u6a21\u3001\u673a\u5668\u5b66\u4e60\u3001\u7edf\u8ba1\u7269\u7406\u3001\u7c7b\u8111\u8ba1\u7b97\u7b49\u591a\u79cd\u6280\u672f\u65b9\u6cd5\u7684\u53d1\u5c55\uff0c\u6211\u4eec\u5bf9\u5927\u8111\u795e\u7ecf\u5143\u76f8\u4e92\u4f5c\u7528\u673a\u7406\u4e0e\u8fde\u63a5\u673a\u5236\uff0c\u5bf9\u610f\u8bc6\u3001\u8bed\u8a00\u3001\u60c5\u7eea\u3001\u8bb0\u5fc6\u3001\u793e\u4ea4\u7b49\u529f\u80fd\u7684\u8ba4\u8bc6\u9010\u6e10\u6df1\u5165\uff0c\u5927\u8111\u590d\u6742\u7cfb\u7edf\u7684\u8c1c\u5e95\u6b63\u5728\u88ab\u63ed\u5f00\u3002<\/span>\u4e3a\u4e86\u4fc3\u8fdb\u795e\u7ecf\u79d1\u5b66\u3001\u7cfb\u7edf\u79d1\u5b66\u3001\u8ba1\u7b97\u673a\u79d1\u5b66\u7b49\u9886\u57df\u7814\u7a76\u8005\u7684\u4ea4\u6d41\u5408\u4f5c\uff0c\u6211\u4eec\u53d1\u8d77\u4e86\u3010<\/span>\u795e\u7ecf\u52a8\u529b\u5b66\u6a21\u578b\u8bfb\u4e66\u4f1a<\/a>\u3011\u3002<\/span><\/p>\n


<\/p>\n

\u96c6\u667a\u4ff1\u4e50\u90e8\u8bfb\u4e66\u4f1a\u662f\u9762\u5411\u5e7f\u5927\u79d1\u7814\u5de5\u4f5c\u8005\u7684\u7cfb\u5217\u8bba\u6587\u7814\u8bfb\u6d3b\u52a8\uff0c\u5176\u76ee\u7684\u662f\u5171\u540c\u6df1\u5165\u5b66\u4e60\u63a2\u8ba8\u67d0\u4e2a\u79d1\u5b66\u8bae\u9898\uff0c\u6fc0\u53d1\u79d1\u7814\u7075\u611f\uff0c\u4fc3\u8fdb\u79d1\u7814\u5408\u4f5c\u3002\u3010\u795e\u7ecf\u52a8\u529b\u5b66\u6a21\u578b\u8bfb\u4e66\u4f1a<\/a>\u3011<\/span>\u7531\u96c6\u667a\u4ff1\u4e50\u90e8\u548c\u5929\u6865\u8111\u79d1\u5b66\u7814\u7a76\u9662\u8054\u5408\u53d1\u8d77\uff0c\u5df2<\/span>\u4e8e3\u670819\u65e5\u5f00\u59cb\uff0c<\/span>\u6bcf\u5468\u516d\u4e0b\u534814:00-16:00\uff08\u6216\u6bcf\u5468\u4e94\u665a\u4e0a19:00-21:00\uff0c\u6839\u636e\u5b9e\u9645\u60c5\u51b5\u8c03\u6574\uff09\u8fdb\u884c\uff0c\u9884\u8ba1\u6301\u7eed10-12\u5468<\/span>\u3002\u671f\u95f4\u5c06\u56f4\u7ed5\u795e\u7ecf\u7f51\u7edc\u591a\u5c3a\u5ea6\u5efa\u6a21\u53ca\u5176\u5728\u8111\u75be\u75c5\u3001\u8111\u8ba4\u77e5\u65b9\u9762\u7684\u5e94\u7528\u8fdb\u884c\u7814\u8ba8\u3002<\/span><\/p>\n

<\/span>
<\/p>\n


<\/section>\n
\n


<\/p>\n<\/section>\n

\u8be6\u60c5\u8bf7\u89c1\uff1a<\/span>
<\/p>\n

\u795e\u7ecf\u52a8\u529b\u5b66\u6a21\u578b\u8bfb\u4e66\u4f1a\u542f\u52a8\uff1a\u6574\u5408\u8ba1\u7b97\u795e\u7ecf\u79d1\u5b66\u7684\u591a\u5b66\u79d1\u65b9\u6cd5<\/a><\/strong><\/p>\n


<\/section>\n

<\/section>\n
\n

\u63a8\u8350\u9605\u8bfb<\/span><\/strong><\/strong><\/p>\n