Efficient data transmission over dynamical complex networks Giacomo Baggio Virginia Rutten Guillaume Hennequin Sandro Zampieri
Richly structured transients in M Motor cortex dynamics: fromet al, Neuro [Churchland et al, Nature (212); Hennequin experiments to dynamical models dx = dt [Churchland et al., Nature (212)] 5 Hz A 1 B 2 3 movement generation movement done go cue x + W g [x(t single neuron rates 2 ms movement preparation The signal generated is characterized by a rich transient and by asymptotic stability target go cue 2
Introduction to neuronal networks Peter Dayan and L.F. Abbott Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems The MIT Press, 21 This is based on a linearized model of the interaction dynamics between neurons The signals are the spiking rates 3
A model for information transmission The question is how populations of neurons produce large amplitude transient signals for information transmission and storage. In various papers it is emphasized the role the system non-normality for the generation of the best signals. 1.S. Ganguli, D. Huh, and H. Sompolinsky Memory traces in dynamical systems. Proceedings of the National Academy of Sciences, vol. 15.48, pp. 1897-18975, 28.S. 2.Ganguli, and P. Latham, Feedforward to the Past: The Relation between Neuronal Connectivity, Amplification, and Short-Term Memory, Neuron, Vol. 41, pp. 499-51, 29. 3.M. S. Goldman, Memory without Feedback in a Neural Network, Neuron, Vol. 61, pp. 621 634, 29. 4.G. Hennequin, T. P. Vogels, and W. Gerstner, Non-normal amplification in random balanced neuronal networks, Physical Review E, Vol. 86, pp. 1199, 212. 5.G. Hennequin, et al. Optimal control of transient dynamics in balanced networks supports generation of complex movements. Neuron, 82.6 (214): 1394-146. The key point is to have large transients while keeping the dynamics stable 4
A model for information transmission Problems: 1. Propose a consistent model for understanding why nonnormality plays a role in making information transmission more efficient. 2. Quantify the information transmission efficiency. 3. Verify whether the model complexity influences information transmission efficiency. 5
Generation of rich signals It is important to produce transients that have high energy initially and then converge to zero fast 6
Channel capacity Channel model: conditional probability p(z Y ) Y channel Z Shannon channel capacity C = max p(y ) I(Y ; Z) = max p(y ) h(z) h(z Y ) where I( ) is the mutual information and h( ) is the di erential entropy 7
Continuous time channels Encoder + Modulator + Demodulator + Decoder T T The capacity depends on the signal to noise ratio SNR 8
Modulation by a system transient ẋ(t) =Ax(t)+Bu(t) y(t) =Cx(t) 9
Modulation by a system transient At time t = with a symbol a 2 A we associate an impulsive input u(t) =u (t) with ku kapplep ẋ(t) =Ax(t)+Bu(t) y(t) =Cx(t) n(t) y 1 y y 1 a 1 a T a 1 t Encoder + Modulator y 1 y y 1 t t + ỹ 1 ỹ ỹ 1 T T y k (t) =Ce A(t t kt ) Bu k 1
Inter-symbol interference T 11
Inter-symbol interference y 1 (t) =Ce A(t T ) Bu 1 -T T Inter-symbol interference: The signal transmitted at time -T, -2T, -3T,... interfere with the signal transmitted at time and acts as an additional noise. 12
Expression of the capacity R T = 1 2T max,tr applep log 2 det ( I + OW) det ( I + O(W B B > )), where O := Z T e A>t C > Ce At dt denotes the [, T]-observability Gramian of the pair (A, C) and W := 1X k= e AkT B B > e A> kt denotes the discrete-time controllability Gramian of the pair (e AT,B 1/2 ). 13
Expression of the capacity R T = 1 2T max,tr applep log 2 det ( I + OW) det ( I + O(W B B > )), where O := Z T e A>t C > Ce At dt denotes the [, T]-observability Gramian of the pair (A, C) and W := 1X k= e AkT B B > e A> kt denotes the discrete-time controllability Gramian of the pair (e AT,B 1/2 ). R max := max R T T <latexit sha1_base64="otvx5yvitbczvuyo59xayyad9s=">aaab2hiczu/lssnafl2prza+oi7dbnucq5jeurakbtcua+kltqmt6bqdopoezfrbqscd6nk9/+bw/8s/malbbhvgcs+ccy7mcqnghtsmbyw3tr6xuzuvqns7u3v72sfhw/itejmw9pkfdlkckmeaukqgekgiuhczatjjq9tv/naqkf9rylnaelxnptoggike8nrsmqppbyatmrzni2vqulyoqy9jlorn5xmyuuqoxwnijghvkrmjbqhq93r3u2+jyecebizjmt9rsb7eqolxyzeqi3jvd7svhxvtqnztfxpoe/e9ksqaoexgpjoxjhwy4nu1wd+miwn9bm6kencibl3kyrhciswvvt885i+5vlvvnbqphnfevwktasrfkejueetsges6jbddshbrhe4qm+4uu5u56uz+xln5ptspsjwidy9gnljih</latexit> 14
Matrix non-normality A matrix is normal i AA T = A T A <latexit sha1_base64="pi/x35aa3sgouwlmpnyo2zucvo=">aaab13iczu/ltgjbeoz1ifhcpxqzccseyo5q9etcxoththgzqti7zmkemd3nzkaqqrwzpfobfonx/rt/xge3rqcstqqrqpmup+zmadv+tlzw19y3nlnb6e2d3b39zmfhtuudswivrdysdr8rylliq5ppthuxpfj4nnb9/txurz9qqvguvvqopi2buyelgmhaso1m1kmca8mgickurljgjlgqojzn3vekzpcxmym7ym+alomtkcwkklcz781oraachppwrntdraxbyyw1i5xok1nh/qrdxsv6nheppp/hhoznwekxpjczeozwnkg+kdgoiasbuakbo5bbqair8kzr1emrrm4p/nunjlh6/tgpuwtkruddutuqmzvjwdcdwcg5cqgmuoqxvipakh/ajx9at9wq9wy+/rurutmcovhvpx5zgpo=</latexit> 15
Matrix non-normality Non-normality has two important features: 1. The eigenvalues of a highly non-normal matrix A are very sensitive to matrix entries variations. 2. The exponential e At of a stable highly non-normal matrix A is large for small t and then decays to zero according to the spectral abscissa (A). non-normality dominant eigenvalue (A) 16
The normal case For normal networks R T is decreasing in T and so the smaller is T the better the performance we obtain. Corollary If V in = V out = V and A 2 R n n is a normal and stable matrix, then R max := max T R T = R = 1 ln 2 tr(a) SNR SNR 2tr(A). 17
<latexit sha1_base64="v8yahia3tid5yzrgfxrckc1gblk=">aaabyhicdy9lt8jafixv4avxvr87n41a4skqtiissejcgfeyycoxheynf5gwzadqutcxv9h4lb/kf/ggmjaxjnmcuy75y6ofwmutgv9ktta+sbmvno7s7o7t39ghb41vdimgdzzkmk45vgfggdy11wlbeuxuukjbhqd61nefmry8tc41+mi25l2at7ljooedyytnoujlrydegaroejs3mdi2sxrllma8kuy1apzp5ewvio1jfext9kq4mbzoiq9vckdhtcy82zwgng1fikr9zx/yptmznjl4hl5o8ofuaudwgpj/njuzofin/shnhyam3o6uqpsqxgkuakuq++pvn4g+2vod/3akdrhg3dnzqrnyloztoinzskemvbibgtsbwto8wtt8kfsskrezf1dtzhfzdcsil18luhs</latexit> <latexit sha1_base64="ajdgqhgag6dp7yufhyktaqzoaua=">aaabu3iczy9lt8jafivv8yx4ql26aqqsn5k2gf2rklhxiyk8jcvkor3ayezbdg5vqvgvbuwh+w8s2bibkxy5jtnkjlejiumy/o2clvbo7t7+f3cwehr8unx9kytwytm2okhdoouxzrkewclbensrjey5unseop7rd55w1ilmhiisyq9xyabgajokexp5wvxrms3c+wrkq1lllp7myuifozx5y7fsgthqfxybr+uy2on2uxcs5xvnajp+hd+dsq2xzxhco/cjpe3urjxpiydxg6xdezkynyzueypycgclxekxppvfe2lsmrno9w8c/ln1jr/9+7sdql2roo9oqefky/jwazdwbtbcqqmeoakt4kdge75gbtqnbrwa8ream7i357aii/kbxnv5iw==</latexit> <latexit sha1_base64="lqdg8ubivwfzwb5rpp8pzyw=">aaabuxiczy9dswjbgixfss+zr6uu1lsosvzxamuihc66digp6avmr1fdxjmd9l5rut8edw/bd+tastkxpg4mxzzsccifbskon8s9zg5tb2tn63sld/chhkhz8ttrobdzepkkkhxcdsobyiekk23gcxackw8hobp63xjaxmgofarjjr/nbkptsceprq+qhslzutypoxvnixjduzoqqqd61vvxejmyaqxkkg/nfvnnyhosqugs4bo+avsfdwdyqulp+by/tujw3gxiz4akeldto7nkke3y+s9irkl+hsj2tjjjpim5rtkxmc/ixpxvc1d+vm6zxcasv78er1rxswr7o4bwuwivrqme91kebakbwdh/wyw4yzp2/fvnseznksyjmr9ehxh</latexit> <latexit sha1_base64="lqdg8ubivwfzwb5rpp8pzyw=">aaabuxiczy9dswjbgixfss+zr6uu1lsosvzxamuihc66digp6avmr1fdxjmd9l5rut8edw/bd+tastkxpg4mxzzsccifbskon8s9zg5tb2tn63sld/chhkhz8ttrobdzepkkkhxcdsobyiekk23gcxackw8hobp63xjaxmgofarjjr/nbkptsceprq+qhslzutypoxvnixjduzoqqqd61vvxejmyaqxkkg/nfvnnyhosqugs4bo+avsfdwdyqulp+by/tujw3gxiz4akeldto7nkke3y+s9irkl+hsj2tjjjpim5rtkxmc/ixpxvc1d+vm6zxcasv78er1rxswr7o4bwuwivrqme91kebakbwdh/wyw4yzp2/fvnseznksyjmr9ehxh</latexit> <latexit sha1_base64="jarao/t1cph+35qnvenk1i/zk=">aaabtxiczy9lt8jafixv4avxhbpwgkrkhbjaxmsny4xcipraizti8wyabtdc4qifwet7ryh/lvlngywznmcuy75yb3brgsllz3i+u2nre2d/k7hb39g8oj4vfjy5pjirapjdjjj+awlyywsziuduieuq4utopx7sjvp2nipykearpjt/nhjadscerrqzkq94slt+ou5fw3xmzkkknrl352qymmgimsilv7db1tb8ytkklhvnalfkuxgdloxnofllt+gkv351yjlky8yholvvpnuqkqmdgkvrf5czpso9ra6c6sjua8iuzwv4m6x3eovb/zctv+pdvv17v1t3s8vycabncaee1kaod9cajggywhu8wwersr4l2ecnmmpzzcmsijlvp/x2wg==</latexit> Example Im( ).5 =5, = 4, =1 =1.5 8 6 4 2 Re( ) /... 1 2 3 NN n = 1, N 1, r 2 2 =1, P =1 <latexit sha1_base64="wf3veoejaubfnbjfcbzxwi8m3ui=">aaabzxiczu/lsgmxfl2pr1pfo67ezwbbccflmoqucgu37qxgh+dukkntnjszgsyztdtqq9w6va/x78xrypy9scfc885iff4kebko843yiwtr6yuzddzg5tb2zvw7l5dhulmwy2gioybplfm8idvnnecnaoyeekl1vahlxo/8cbixcpgvg8j1pkkf/aupqbqwdfiiydk4821o8j8m9w8zmqzzxow3lnzizhb1icerykklatt69tkgtyqjnbvhq7jzsrrgjnaecjxoezk/6kxd33xiza74tzgzu5cofhe6id2mvya2ujdexugjsjtd1vz3jekjwuvklkovtq3puif57pg+evxyr1t4rps+6nm6+4abmshmirhaogc6jafvshbhre4qm+4qtdowq9o5ffaaalb/zhbujtb5a9fey=</latexit> = 1, P =1 RT 1.8.6.4.2... n.5 1 1.5 2 T 18
<latexit sha1_base64="ra7ezsg9rbo4kvzhqumv62hkk=">aaab3icdy/ntsjafixv4b/ix9wlcxubxiuhbuvdkzc4cymjbrnlyhr6gqkzbdmzvelygf36cd6bw3y38acaedjssy5851zf8epbvfasj5izmfxaxklu5pbw9/y3dk2d+oq6icmxrajkln2qulbq3q11wkv4wsp9au2/n75og/cyqj4ff7pqyxnstshb3ngdypaxn7bc1bowikfeabny+rssami7tlksafl5o2snzfp/thfur6mqrwmfy+iwf9iqjmgst2cxro5pinmtoao52m813c8n2kbtgzk86acvr3+gpjynqg8pjupfztffgtqmkfae2j3sur6vsa+mntul1v/3oxvanm93zv6k7jfu45fw6+aozxzafptiaq7dhdkpwatvwgcetvmibvboxdmkdefyqzsjzhfmrj4/awjgp8=</latexit> <latexit sha1_base64="whu/yqcovcfhlgtdnliyqbzayv8=">aaabv3iczy9dswjbgixfss+zl6vlbpzu6ep2t6irwoims4nmorwzhv91cgz32nmtrpwdxqx1s/o3rbze6ogbm885a3nco6ql1/1mubx1jc2t/hzhz3dv/6b4epro41eiscfifsetkftumsigsvlymglyhspshspbwd58wctkohqgscg25v1i9qtglkj2oqiremcvgfbyp1hyq+5czqrxmloctpvo8spoxmkkmskhulvpl4bae56qfaqnhydwjv5llwbxnitofipxkt3ygtrcdhkfzzmlydsoq6ti9ohorm6clpa6theswbwpoa7uczebflu7xln+/ah79qnde9e/9us3pluxhbe7hddy4ghrcqraioaz3uetvtgn67oimd9qjmvvjmfbbpwdvgv7eg==</latexit> Example: Non normal matrix Im( ) / / = =4, 4, =1, = =5.5.5 8 6 4 2 Re( )...... 1 2 3 N 1 2 3 N / n <latexit sha1_base64="jarao/t1cph+35qnvenk1i/zk=">aaabtxiczy9lt8jafixv4avxhbpwgkrkhbjaxmsny4xcipraizti8wyabtdc4qifwet7ryh/lvlngywznmcuy75yb3brgsllz3i+u2nre2d/k7hb39g8oj4vfjy5pjirapjdjjj+awlyywsziuduieuq4utopx7sjvp2nipykearpjt/nhjadscerrqzkq94slt+ou5fw3xmzkkknrl352qymmgimsilv7db1tb8ytkklhvnalfkuxgdloxnofllt+gkv351yjlky8yholvvpnuqkqmdgkvrf5czpso9ra6c6sjua8iuzwv4m6x3eovb/zctv+pdvv17v1t3s8vycabncaee1kaod9cajggywhu8wwersr4l2ecnmmpzzcmsijlvp/x2wg==</latexit> N n = 1, 1, r 2 = 2 =1, P =1 P =1 <latexit sha1_base64="wf3veoejaubfnbjfcbzxwi8m3ui=">aaabzxiczu/lsgmxfl2pr1pfo67ezwbbccflmoqucgu37qxgh+dukkntnjszgsyztdtqq9w6va/x78xrypy9scfc885iff4kebko843yiwtr6yuzddzg5tb2zvw7l5dhulmwy2gioybplfm8idvnnecnaoyeekl1vahlxo/8cbixcpgvg8j1pkkf/aupqbqwdfiiydk4821o8j8m9w8zmqzzxow3lnzizhb1icerykklatt69tkgtyqjnbvhq7jzsrrgjnaecjxoezk/6kxd33xiza74tzgzu5cofhe6id2mvya2ujdexugjsjtd1vz3jekjwuvklkovtq3puif57pg+evxyr1t4rps+6nm6+4abmshmirhaogc6jafvshbhre4qm+4qtdowq9o5ffaaalb/zhbujtb5a9fey=</latexit> RT 1.25 1.75.5.25 2 4 6 8 T 19
<latexit sha1_base64="zg7dusb4jacveuofem7dt/4jrne=">aaab3iczy/dsgjbhmx/a19mx1tddtgscl2e7g6wecei3xrpkrqiszo/txb2q92xkrem6jlhqen6lyeprdpni3mawnnfucmzpfizoqzs8ttbk6tr6r3sxsbe/s7un7b3urdrkkdo14ldq9ipczeb3jjmdmncajpi4nr381yrv3magwhbdyggmrin2qdrgluqg2fpxzfessvipnruf6qjw1cythpvip5dp61iyyuxnklg27fgfyczi3wzip1tbfxd+igwbdstkr4u4ylqrsssjhmczv+kjfgc+7fuskwaz/b9qvhd3idamte+6ojqugxt5hxyjeyxqhnkyouecyqybp5qbkt2xp9san8ztwfp98umbhes84j9y2er9mxzgo7gbe7bghju4rpq4acff3ihd/juhg2kpwnpp9wunntzcavsxr8bksgbha==</latexit> <latexit sha1_base64="ra7ezsg9rbo4kvzhqumv62hkk=">aaab3icdy/ntsjafixv4b/ix9wlcxubxiuhbuvdkzc4cymjbrnlyhr6gqkzbdmzvelygf36cd6bw3y38acaedjssy5851zf8epbvfasj5izmfxaxklu5pbw9/y3dk2d+oq6icmxrajkln2qulbq3q11wkv4wsp9au2/n75og/cyqj4ff7pqyxnstshb3ngdypaxn7bc1bowikfeabny+rssami7tlksafl5o2snzfp/thfur6mqrwmfy+iwf9iqjmgst2cxro5pinmtoao52m813c8n2kbtgzk86acvr3+gpjynqg8pjupfztffgtqmkfae2j3sur6vsa+mntul1v/3oxvanm93zv6k7jfu45fw6+aozxzafptiaq7dhdkpwatvwgcetvmibvboxdmkdefyqzsjzhfmrj4/awjgp8=</latexit> <latexit sha1_base64="whu/yqcovcfhlgtdnliyqbzayv8=">aaabv3iczy9dswjbgixfss+zl6vlbpzu6ep2t6irwoims4nmorwzhv91cgz32nmtrpwdxqx1s/o3rbze6ogbm885a3nco6ql1/1mubx1jc2t/hzhz3dv/6b4epro41eiscfifsetkftumsigsvlymglyhspshspbwd58wctkohqgscg25v1i9qtglkj2oqiremcvgfbyp1hyq+5czqrxmloctpvo8spoxmkkmskhulvpl4bae56qfaqnhydwjv5llwbxnitofipxkt3ygtrcdhkfzzmlydsoq6ti9ohorm6clpa6theswbwpoa7uczebflu7xln+/ah79qnde9e/9us3pluxhbe7hddy4ghrcqraioaz3uetvtgn67oimd9qjmvvjmfbbpwdvgv7eg==</latexit> Example: Non normal matrix Im( ) = 4, = 1, =5.5.5 8 6 4 2 Re( ) = 4, = 1, =7 / =4, =1, =5 =4, =1, =7... 1 / 2 3 N N = 1, 2 r = 1, P =1 RT 3 2.5 2 1.5 1.5... / n <latexit sha1_base64="jarao/t1cph+35qnvenk1i/zk=">aaabtxiczy9lt8jafixv4avxhbpwgkrkhbjaxmsny4xcipraizti8wyabtdc4qifwet7ryh/lvlngywznmcuy75yb3brgsllz3i+u2nre2d/k7hb39g8oj4vfjy5pjirapjdjjj+awlyywsziuduieuq4utopx7sjvp2nipykearpjt/nhjadscerrqzkq94slt+ou5fw3xmzkkknrl352qymmgimsilv7db1tb8ytkklhvnalfkuxgdloxnofllt+gkv351yjlky8yholvvpnuqkqmdgkvrf5czpso9ra6c6sjua8iuzwv4m6x3eovb/zctv+pdvv17v1t3s8vycabncaee1kaod9cajggywhu8wwersr4l2ecnmmpzzcmsijlvp/x2wg==</latexit> 1 2 3 N n = 1, 2 =1, P =1 <latexit sha1_base64="wf3veoejaubfnbjfcbzxwi8m3ui=">aaabzxiczu/lsgmxfl2pr1pfo67ezwbbccflmoqucgu37qxgh+dukkntnjszgsyztdtqq9w6va/x78xrypy9scfc885iff4kebko843yiwtr6yuzddzg5tb2zvw7l5dhulmwy2gioybplfm8idvnnecnaoyeekl1vahlxo/8cbixcpgvg8j1pkkf/aupqbqwdfiiydk4821o8j8m9w8zmqzzxow3lnzizhb1icerykklatt69tkgtyqjnbvhq7jzsrrgjnaecjxoezk/6kxd33xiza74tzgzu5cofhe6id2mvya2ujdexugjsjtd1vz3jekjwuvklkovtq3puif57pg+evxyr1t4rps+6nm6+4abmshmirhaogc6jafvshbhre4qm+4qtdowq9o5ffaaalb/zhbujtb5a9fey=</latexit> 2 4 6 8 T 2
<latexit sha1_base64="ajdgqhgag6dp7yufhyktaqzoaua=">aaabu3iczy9lt8jafivv8yx4ql26aqqsn5k2gf2rklhxiyk8jcvkor3ayezbdg5vqvgvbuwh+w8s2bibkxy5jtnkjlejiumy/o2clvbo7t7+f3cwehr8unx9kytwytm2okhdoouxzrkewclbensrjey5unseop7rd55w1ilmhiisyq9xyabgajokexp5wvxrms3c+wrkq1lllp7myuifozx5y7fsgthqfxybr+uy2on2uxcs5xvnajp+hd+dsq2xzxhco/cjpe3urjxpiydxg6xdezkynyzueypycgclxekxppvfe2lsmrno9w8c/ln1jr/9+7sdql2roo9oqefky/jwazdwbtbcqqmeoakt4kdge75gbtqnbrwa8ream7i357aii/kbxnv5iw==</latexit> <latexit sha1_base64="myp4xuthiq6cz5xzpis78coeedo=">aaabwhiczy/ltgixgix/4g3xnurszuqgcyuzo9gvcdgnszkkjhiouwhawin47soshgpny7rxwbb5wygzmof3oaditxijr4zjfjleyura+kd8sbg3v7o5z+wcnryyjwzptqiwtggoupmk64uzgk6qykbgmxjctppmmyaaq+jejgjss9qpei8zallwpidnptupyioaaljfz2km5o9bnzmfcftrwn9+f3fhhijwwtv+ueinuxtqxnaicf3+creefde165dpof8j9wnt79ocaysght43g2zwkxu9s1eypjt2tsgz3ruan1sazpu1it6svscv+ydi+7+ptl/aq7lnfu/okvs9blocjoiytcoesqnalnagdgwte4ro+ydujisjpv9ucyd4cwpzi2w+6entl</latexit> <latexit sha1_base64="whu/yqcovcfhlgtdnliyqbzayv8=">aaabv3iczy9dswjbgixfss+zl6vlbpzu6ep2t6irwoims4nmorwzhv91cgz32nmtrpwdxqx1s/o3rbze6ogbm885a3nco6ql1/1mubx1jc2t/hzhz3dv/6b4epro41eiscfifsetkftumsigsvlymglyhspshspbwd58wctkohqgscg25v1i9qtglkj2oqiremcvgfbyp1hyq+5czqrxmloctpvo8spoxmkkmskhulvpl4bae56qfaqnhydwjv5llwbxnitofipxkt3ygtrcdhkfzzmlydsoq6ti9ohorm6clpa6theswbwpoa7uczebflu7xln+/ah79qnde9e/9us3pluxhbe7hddy4ghrcqraioaz3uetvtgn67oimd9qjmvvjmfbbpwdvgv7eg==</latexit> <latexit sha1_base64="jarao/t1cph+35qnvenk1i/zk=">aaabtxiczy9lt8jafixv4avxhbpwgkrkhbjaxmsny4xcipraizti8wyabtdc4qifwet7ryh/lvlngywznmcuy75yb3brgsllz3i+u2nre2d/k7hb39g8oj4vfjy5pjirapjdjjj+awlyywsziuduieuq4utopx7sjvp2nipykearpjt/nhjadscerrqzkq94slt+ou5fw3xmzkkknrl352qymmgimsilv7db1tb8ytkklhvnalfkuxgdloxnofllt+gkv351yjlky8yholvvpnuqkqmdgkvrf5czpso9ra6c6sjua8iuzwv4m6x3eovb/zctv+pdvv17v1t3s8vycabncaee1kaod9cajggywhu8wwersr4l2ecnmmpzzcmsijlvp/x2wg==</latexit> Line network: varying anysotropy Example: chain network strength 1 Im( ) 1 1 1 1 = 4, = 1, =5.5 1 3 1 5.5 8 61 4 6 2 Re( ) = 4, = 1, =7 / / 1 3.5 1 1 3 1 6 5... 1 2 3 N N = 1, 2 r = 1, P =1 RT 3 2.5 2 1.5 1 n 2 4 6 8 5T varying <latexit sha1_base64="n5xte/nsnewcvc2zmsqnbcxpi=">aaabxhiczy9ds8mwgixfzk85v6peelpcbl6ntopecqnbvjzgpsco8tbltrcklulahwp+er3u/+s/sztf3hygcpkce8gjysg1czxvutjy3nreke6w9vypdo+s45owjhjfwzngilkdadutpgrnw41gnvgxlifg7wb8n8/bkvoar+gtmcssk3ey8ggnadlus6wu1yshq7vio4hhwolzzafmlgsvgzc3zcjv6fkffj+iiwshoqk1fr6otxekynaq2kzkg/zqxnjfjg5dh8ps9r+4yu5+olmmdixdnl3mmdnvdpxtqasyexp7qzd6klweycbrsjqjvzrn4v+w7xfxf79uwl7nvax5j1657uxlinag53abltxahr6gau2gkmi7fmixusecajl8vgskf3mksyjvp8prfik=</latexit> 5 2.5 2.5 2 4 6 T 2 4 6 T 2 4 6 T 21
<latexit sha1_base64="ajdgqhgag6dp7yufhyktaqzoaua=">aaabu3iczy9lt8jafivv8yx4ql26aqqsn5k2gf2rklhxiyk8jcvkor3ayezbdg5vqvgvbuwh+w8s2bibkxy5jtnkjlejiumy/o2clvbo7t7+f3cwehr8unx9kytwytm2okhdoouxzrkewclbensrjey5unseop7rd55w1ilmhiisyq9xyabgajokexp5wvxrms3c+wrkq1lllp7myuifozx5y7fsgthqfxybr+uy2on2uxcs5xvnajp+hd+dsq2xzxhco/cjpe3urjxpiydxg6xdezkynyzueypycgclxekxppvfe2lsmrno9w8c/ln1jr/9+7sdql2roo9oqefky/jwazdwbtbcqqmeoakt4kdge75gbtqnbrwa8ream7i357aii/kbxnv5iw==</latexit> <latexit sha1_base64="myp4xuthiq6cz5xzpis78coeedo=">aaabwhiczy/ltgixgix/4g3xnurszuqgcyuzo9gvcdgnszkkjhiouwhawin47soshgpny7rxwbb5wygzmof3oaditxijr4zjfjleyura+kd8sbg3v7o5z+wcnryyjwzptqiwtggoupmk64uzgk6qykbgmxjctppmmyaaq+jejgjss9qpei8zallwpidnptupyioaaljfz2km5o9bnzmfcftrwn9+f3fhhijwwtv+ueinuxtqxnaicf3+creefde165dpof8j9wnt79ocaysght43g2zwkxu9s1eypjt2tsgz3ruan1sazpu1it6svscv+ydi+7+ptl/aq7lnfu/okvs9blocjoiytcoesqnalnagdgwte4ro+ydujisjpv9ucyd4cwpzi2w+6entl</latexit> <latexit sha1_base64="whu/yqcovcfhlgtdnliyqbzayv8=">aaabv3iczy9dswjbgixfss+zl6vlbpzu6ep2t6irwoims4nmorwzhv91cgz32nmtrpwdxqx1s/o3rbze6ogbm885a3nco6ql1/1mubx1jc2t/hzhz3dv/6b4epro41eiscfifsetkftumsigsvlymglyhspshspbwd58wctkohqgscg25v1i9qtglkj2oqiremcvgfbyp1hyq+5czqrxmloctpvo8spoxmkkmskhulvpl4bae56qfaqnhydwjv5llwbxnitofipxkt3ygtrcdhkfzzmlydsoq6ti9ohorm6clpa6theswbwpoa7uczebflu7xln+/ah79qnde9e/9us3pluxhbe7hddy4ghrcqraioaz3uetvtgn67oimd9qjmvvjmfbbpwdvgv7eg==</latexit> <latexit sha1_base64="jarao/t1cph+35qnvenk1i/zk=">aaabtxiczy9lt8jafixv4avxhbpwgkrkhbjaxmsny4xcipraizti8wyabtdc4qifwet7ryh/lvlngywznmcuy75yb3brgsllz3i+u2nre2d/k7hb39g8oj4vfjy5pjirapjdjjj+awlyywsziuduieuq4utopx7sjvp2nipykearpjt/nhjadscerrqzkq94slt+ou5fw3xmzkkknrl352qymmgimsilv7db1tb8ytkklhvnalfkuxgdloxnofllt+gkv351yjlky8yholvvpnuqkqmdgkvrf5czpso9ra6c6sjua8iuzwv4m6x3eovb/zctv+pdvv17v1t3s8vycabncaee1kaod9cajggywhu8wwersr4l2ecnmmpzzcmsijlvp/x2wg==</latexit> Line network: varying length Example: chain network / /... n 1 2 3 N Im( ) 1 1 1 1-1 = 4, = 1, =5.5 1-3 1-5.5 1 8 6 4-1 6 2-1 3.5 1 1 3 1 6 Re( ) = 4, = 1, =7 N = 1, 2 r = 1, P =1 4 RT 3 2.5 2 1.5 2 4 6 8 T4 varying n <latexit sha1_base64="fcfg/kjqxxx27gjxwbnyizlwy6q=">aaabvxiczy9bswjbhmx/yzezm9vjlsq9cs7w9rtjptsofeqevmx786ojdlztre/bg9vp+rb9nqs6qdgdjzo2dgthqlbp3vf5hm1vbo7l52p3dwehr8kj89q1s9ngxrtattmhg1kljcmunoydm2sguksbgnhpd5y4lgcq1e3czgjqqdxfucuzeg1osagvcdr6ik3xzbl/sref9nkjocpkp28+/tnmzjicoxqa1t3caum6fgcszwkws7fhvt3npd+8bnmlf6a26se3tsmazsrac4x+1yekue9by+nslrzlvrtr6v1s5kldqlduo7ms3hb5bsctz//9/uw3jwxq6fwiltjdl4qiu4qocuimkpeevasbawxt8wcd5iegeut/vdenfnmoaypqb+nn6mg==</latexit> 5 2 2 2 4 6 T 2 4 6 T 2 4 6 T 22
<latexit sha1_base64="7eqtpskxlrf2iw5qxbtlgmaule=">aaacgniczvdlssqwfe19w1+jltep4kcdmvxqmkg92pocoyhdlm7uyyjq1jxgelf6i/456cbfckslm+mgpg6ee+45j3dvjbkea+p7r5f/8dgpdiqds2pje5vzqeodzpwzgosjrj1wlensrcqtvwk8bppockkigtqlxt8u+uqgmeyinzm8g5oa3jy86osvktfbozclkhady9gjsmyntrwfducsozyi9o3hdiiqmxb633mli1/moazkr9kawtoblvejirvkefdljja5tuqibvsxul3kedmulsl/xu8d/sdajzdtdfq1t+opaws7geuo1mfrmtnpqtkcjvc4xmcnuez19wmfcbchr/cmu1jwngwys2io8eqcalvqrjofx2syo76q8cfvrfisgmbtvn/dfrin+e3sf4o/1/chxwgtvkebcwt8lezinods2jrrsgdbsfdte+qikgntebekcfzp3z4dw6t1/rpqf3zxb9gvpycqatnmk=</latexit> <latexit sha1_base64="a7otbfsskub/isx8najats+mxq=">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</latexit> Limit behaviors in the noise If the noise variance 2!, then R max ' 1 ln 2 tr(a) If the noise variance 2!1,then R max ' 1 2ln2 ` 2 where ` := max T kb T O T Bk T O T = Z T e AT t C T Ce At dt 23
<latexit sha1_base64="tnalsrhykny7pmuqecmggk+zps=">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</latexit> Matrix non-normality For normal matrices A we have that 1. A is diagonizable by an orthonormal matrix. 2. The eigenvalues of A+AT 2 are the real values of the eigenvalues of A and so!(a) := max{ i : i eigenvalues of A + AT 2 = max{re[ i] : i eigenvalues of A} } 24
<latexit sha1_base64="nknm8amfr2roi2lsktgpa7aqxgu=">aaacdhicbvbbb9mwfhbcbyrb2r7hwaiuklivjqwxp7r+skkpqydppwqozktlxmsxs7bfoun8j/4w/wr3jgzspen45s6fn3fe5vjdcuit5fyt37j94+gjrcftk6bpnl1ovtyfwlmsmxpmrpjzjziiugszooalnrqlmcqmn/gkk+/q2mfsfuuobzxezazetgnkemrz9ug6fza5xlm/wyqjmcaqmnplncbwmqyfpoq5befvl/pbpaktcizwncecmmpwmd8pesprmj8kz2cyukngjnrdt9rlyekh9cdeqibsm7w95bb9qoksgb9j1vxkznmbwmropd/wb3mlgbbwtogkk3wxpgrtruo6nrr8n9ls+zezyaz9+rfw5xurncgk1bf1couure4wgztjvnt5h/jg33rpowdzbztd1ay/xh1p5xhmsn/9hht2w8eutdeke+dqi/a2til/an6e9pbv74jjp7fx/3p/fz+fz3zfnqf3qausteu2kefdeaoyx4gxwf42as/a5fh+2wc2mng3vmb21ugp8bssovna==</latexit> Phase transition Theorem Let A 2 R n n be a stable matrix. If!(A) apple, then `(A, B, C) apple 1 If!(A) > and B = C = I, then `(A, B, C) > 1 25
Line network... / / n <latexit sha1_base64="jarao/t1cph+35qnvenk1i/zk=">aaabtxiczy9lt8jafixv4avxhbpwgkrkhbjaxmsny4xcipraizti8wyabtdc4qifwet7ryh/lvlngywznmcuy75yb3brgsllz3i+u2nre2d/k7hb39g8oj4vfjy5pjirapjdjjj+awlyywsziuduieuq4utopx7sjvp2nipykearpjt/nhjadscerrqzkq94slt+ou5fw3xmzkkknrl352qymmgimsilv7db1tb8ytkklhvnalfkuxgdloxnofllt+gkv351yjlky8yholvvpnuqkqmdgkvrf5czpso9ra6c6sjua8iuzwv4m6x3eovb/zctv+pdvv17v1t3s8vycabncaee1kaod9cajggywhu8wwersr4l2ecnmmpzzcmsijlvp/x2wg==</latexit> 1 2 3 N 1 1 alpha=2.25 alpha=2.5 alpha=2.75 alpha=3 alpha=3.25 1 1-1 1-2 1-3 1-4 1-4 1-2 1 1 2 1 4 26
Line network alpha=2.25 alpha=2.5 alpha=2.75 alpha=3 alpha=3.25 1-2 1 1 2 1 4 27
Line network alpha=2.25 alpha=2.5 alpha=2.75 alpha=3 alpha=3.25 1-2 1 1 2 1 4 28
Line network 1 2 gamma=-6 delta=6 gamma=-5 delta=5 1 1 gamma=-3 delta=3... / / n <latexit sha1_base64="jarao/t1cph+35qnvenk1i/zk=">aaabtxiczy9lt8jafixv4avxhbpwgkrkhbjaxmsny4xcipraizti8wyabtdc4qifwet7ryh/lvlngywznmcuy75yb3brgsllz3i+u2nre2d/k7hb39g8oj4vfjy5pjirapjdjjj+awlyywsziuduieuq4utopx7sjvp2nipykearpjt/nhjadscerrqzkq94slt+ou5fw3xmzkkknrl352qymmgimsilv7db1tb8ytkklhvnalfkuxgdloxnofllt+gkv351yjlky8yholvvpnuqkqmdgkvrf5czpso9ra6c6sjua8iuzwv4m6x3eovb/zctv+pdvv17v1t3s8vycabncaee1kaod9cajggywhu8wwersr4l2ecnmmpzzcmsijlvp/x2wg==</latexit> 1 2 3 N 1 1-1 1-2 1-3 1-4 1-5 1-6 1-6 1-4 1-2 1 1 2 1 4 1 6 29
Line network delta=6 gamma=-6 gamma=-5 delta=5 gamma=-3 delta=3 1-4 1-2 1 1 2 1 4 1 6 3
Line network gamma=-6 gamma=-5 gamma=-3 delta=6 delta=5 delta=3 1-4 1-2 1 1 2 1 4 1 6 31
<latexit sha1_base64="gaqgkdagj4dbqyanhfoj/tugx4=">aaaeyniczvpnbtnaehabai35ayk3okyoe6wqrmttgytuqmk4wauvrnpkbinw9srzdb12vzvsyvkj8ag8ehcehi2ztrn3jmsz3/fn7mys7aecsyxx35xvwv3e/qdrdfvho8dpnq5vpduwysqlybakpmlofsqbmwedxrshzqdgvsctvzz/pq/uyrmskr8v9cpnmuemzeaqonnxy/un8ijjikweref3yjd6eqfyyqoxdawihhmavrtnriuj5oqymwajniqwxegq3g7fkindtr2uktugyjkqaraa+o6gitkijpteaoag5kjtb946iswtu789y24qeof87i7dokyuwiueeiupvlf1dt+tuu3mk6i2ktuovw1gv4s/jpi6tggetcjcamkywp51ni2i7zbzitmf4vrliylsgoofu2hzftrhaxue49pdesqbqngsq6fvx2vqhhyqxrltd255lm3nfzfbhicpcjbmtk+p9geugaeizkhsarjbzx7xrnw2bdhv9zqhnf52iqw69at8t22j3zcffjfpnkssvykrsewi1zsdlwyuzqcazi8zq+2eum71ieund9s3cxawhu45rnc3l2nfw/bdefjcjqaiauyl/vevvwu4zvscourick/wlhwq85+igtpslwk6oyurcsonzgkfe/mwfamoorkmk49qqesxddsw8jr2tvjf9fhwusk45/q8brx7u86x13xfdr2v3tabzyzbs15yr6y25vrvrqprk3vkdayg9rz2odavfaz9q9v1jfrmtlq6yni2rswrv/wpzgzphg==</latexit> Example A = 2 6 4 3................ 7.. 5,B = 2 3 1 6. 7 45,C =... 1, If the noise is small then R max ' n ln 2 Notice moreover that!(a) ' + and so > guarantees that!(a) >. `(A, B, C) 1 4(2n 1) p (n 1) 2n 2 32
Conclusions 1. We proposed a model to quantify the information transmission performance in linear dynamic networks. 2. By introducing the intersymbol interference, we can highlight the role of non-normality of the dynamics. 3. For normal dynamics a theoretical analysis is possible. However this is the less interesting case. 4. Non-normal dynamics is more interesting and harder to study. 33
Thank you 34
Matrix non-normality normal matrices AA = A A A = U ú DU, U unitary, D diagonal (spectral decomposition) A œ R N N non-normal matrices AA = A A A = U ú TU, U unitary, T triangular (Schur decomposition) D = S W U 1 2... N T X V T = S W U 1 ı 2 ı ı... T X V ı ı ı N (A) ={ i } N i=1 T = D + N 35