1932

Abstract

Life depends as much on the flow of information as on the flow of energy. Here we review the many efforts to make this intuition precise. Starting with the building blocks of information theory, we explore examples where it has been possible to measure, directly, the flow of information in biological networks, or more generally where information-theoretic ideas have been used to guide the analysis of experiments. Systems of interest range from single molecules (the sequence diversity in families of proteins) to groups of organisms (the distribution of velocities in flocks of birds), and all scales in between. Many of these analyses are motivated by the idea that biological systems may have evolved to optimize the gathering and representation of information, and we review the experimental evidence for this optimization, again across a wide range of scales.

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2016-03-10
2024-06-20
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Literature Cited

  1. Bialek W. 1.  2012. Biophysics: Searching for Principles. Princeton, NJ: Princeton Univ. Press [Google Scholar]
  2. Shannon CE. 2.  1948. Bell Syst. Tech. J. 27:379–423, 623–56 [Google Scholar]
  3. Cover TM, Thomas JA. 3.  1991. Elements of Information Theory New York: Wiley [Google Scholar]
  4. Mezard M, Montanari A. 4.  2009. Information, Physics, and Computation Oxford/New York: Oxford Univ. Press [Google Scholar]
  5. Kullback S. 5.  1968. Information Theory and Statistics. New York: Dover [Google Scholar]
  6. Berger T. 6.  1971. Rate–Distortion Theory: A Mathematical Basis for Data Compression Englewood Cliffs, NJ: Prentice-Hall [Google Scholar]
  7. Shannon CE. 7.  1959. IRE Natl. Conv. Rec. 4:142–63 [Google Scholar]
  8. Bialek W, Rieke F, de Ruyter van Steveninck RR, Warland D. 8.  1991. Science 252:1854–57 [Google Scholar]
  9. Marre O, Soler VB, Simmons KD, Mora T, Tkačik G, Berry MJ II. 9.  2015. PLOS Comput. Biol. 11:e1004304 [Google Scholar]
  10. Schwartz AB. 10.  1994. Science 265:540–42 [Google Scholar]
  11. Rieke F, Warland D, de Ruyter van Steveninck RR, Bialek W. 11.  1997. Spikes: Exploring the Neural Code. Cambridge, MA: MIT Press [Google Scholar]
  12. Miller GA. 12.  1955. Note on the bias of information estimates. Information Theory in Psychology: Problems and Methods, II-B H Quastler 95–100 Glencoe, IL: Free [Google Scholar]
  13. Panzeri S, Treves A. 13.  1995. Neural Comput. 7:399–407 [Google Scholar]
  14. Strong SP, Koberle R, de Ruyter van Steveninck RR, Bialek W. 14.  1998. Phys. Rev. Lett. 80:197–200 [Google Scholar]
  15. Paninski L. 15.  2003. Neural Comput. 15:1191–253 [Google Scholar]
  16. Nemenman I, Shafee F, Bialek W. 16.  2002. Entropy and inference, revisited. Advances in Neural Information Processing Systems 14 TG Dietterich, S Becker, Z Ghahramani 471–78 Cambridge, MA: MIT Press [Google Scholar]
  17. Nemenman I, Bialek W, de Ruyter van Steveninck RR. 17.  2004. Phys. Rev. E 69:056111 [Google Scholar]
  18. Archer E, Park I, Pillow JW. 18.  2012. Bayesian estimation of discrete entropy with mixtures of stick-breaking priors. Advances in Neural Information Processing Systems 25 F Pereira, CJC Burges, L Bottou, KQ Weinberger 2024–32 Cambridge, MA: MIT Press [Google Scholar]
  19. Archer E, Park I, Pillow JW. 19.  2013. Bayesian entropy estimation for binary spike train data using parametric prior knowledge. Advances in Neural Information Processing Systems 26 CJC Burges, L Bottou, M Welling, Z Ghahramani, KQ Weinberger 1700–8 Cambridge, MA: MIT Press [Google Scholar]
  20. Victor JD. 20.  2002. Phys. Rev. E 66:51903 [Google Scholar]
  21. Kinney JB, Atwal GS. 21.  2014. PNAS 111:3354–59 [Google Scholar]
  22. Ma S-K. 22.  1981. J. Stat. Phys. 26:221–40 [Google Scholar]
  23. Berg OG, von Hippel PH. 23.  1987. J. Mol. Biol. 193:723–50 [Google Scholar]
  24. Schneider TD, Stephens RM. 24.  1990. Nucleic Acids Res. 18:6097–100 [Google Scholar]
  25. Wunderlich Z, Mirny LA. 25.  2009. Trends Genet. 25:434–40 [Google Scholar]
  26. Adrian ED. 26.  1928. The Basis of Sensation: The Action of the Sense Organs New York: W.W. Norton [Google Scholar]
  27. Dubuis JO, Samanta R, Gregor T. 27.  2013. Mol. Syst. Biol. 9:639 [Google Scholar]
  28. Tsien RY. 28.  1998. Annu. Rev. Biochem. 67:509–44 [Google Scholar]
  29. Gregor T, Wieschaus EF, McGregor AP, Bialek W, Tank DW. 29.  2007. Cell 130:141–52 [Google Scholar]
  30. Morrison AH, Scheeler M, Dubuis J, Gregor T. 30.  2012. Quantifying the Bicoid morphogen gradient in living fly embryos. Imaging in Developmental Biology J Sharpe, RO Wang 398–406 Cold Spring Harbor, NY: Cold Spring Harbor Lab. [Google Scholar]
  31. Raj A, van den Bogaard P, Rifkin SA, van Oudenaarden A, Tyagi S. 31.  2008. Nat. Methods 5:877–79 [Google Scholar]
  32. Little SC, Tikhonov M, Gregor T. 32.  2013. Cell 154:789–800 [Google Scholar]
  33. Cheong R, Rhee A, Wang CJ, Nemenman I, Levchenko A. 33.  2011. Science 334:354–58 [Google Scholar]
  34. Selimkhanov J, Taylor B, Yao J, Pilko A, Albeck J. 34.  et al. 2014. Science 346:1370–73 [Google Scholar]
  35. Slonim N, Atwal GS, Tkačik G, Bialek W. 35.  2005. PNAS 102:18297–302 [Google Scholar]
  36. Margolin AA, Nemenman I, Basso K, Wiggins C, Stolovitzky G. 36.  et al. 2006. BMC Bioinform. 7:Suppl. 1S7 [Google Scholar]
  37. Bowers PM, Cokus SJ, Eisenberg D, Yeates TO. 37.  2004. Science 306:2246–49 [Google Scholar]
  38. Slonim N, Elemento O, Tavazoie S. 38.  2006. Mol. Syst. Biol. 2:2006.0005 [Google Scholar]
  39. Hopfield JJ. 39.  1982. PNAS 79:2554–58 [Google Scholar]
  40. Amit DJ. 40.  1989. Modeling Brain Function: The World of Attractor Neural Networks Cambridge, UK: Cambridge Univ. Press [Google Scholar]
  41. Hertz J, Krogh A, Palmer RG. 41.  1991. Introduction to the Theory of Neural Computation Redwood City, CA: Addison-Wesley [Google Scholar]
  42. Toner J, Tu Y. 42.  1995. Phys. Rev. Lett. 75:4326–29 [Google Scholar]
  43. Toner J, Tu Y. 43.  1998. Phys. Rev. E 58:4828–58 [Google Scholar]
  44. Ramaswamy S. 44.  2010. Annu. Rev. Condens. Matter Phys. 1:323–45 [Google Scholar]
  45. Jaynes ET. 45.  1957. Phys. Rev. 106:620–30 [Google Scholar]
  46. Schneidman E, Still S, Berry MJ II, Bialek W. 46.  2003. Phys. Rev. Lett. 91:238701 [Google Scholar]
  47. Keller JB, Zumino B. 47.  1959. J. Chem. Phys. 30:1351–53 [Google Scholar]
  48. Chayes JT, Chayes L, Lieb E. 48.  1984. Commun. Math. Phys. 93:57–121 [Google Scholar]
  49. Caglioti E, Kuna T, Lebowitz J, Speer E. 49.  2006. J. Markov Process. Relat. Fields 12:257–72 [Google Scholar]
  50. Tishby N, Levin E, Solla SA. 50.  1990. Proc. IEEE 78:1568–80 [Google Scholar]
  51. Yedidia JS, Freeman WT, Weiss Y. 51.  2003. Understanding belief propagation and its generalizations. Exploring Artificial Intelligence in the New Millennium G Lakemeyer, B Nebel 239–69 San Francisco, CA: Morgan Kaufmann [Google Scholar]
  52. Mézard M. 52.  2003. Science 301:1685–86 [Google Scholar]
  53. Schneidman E, Berry MJ II, Segev R, Bialek W. 53.  2006. Nature 440:1007–12 [Google Scholar]
  54. Shlens J, Field GD, Gaulthier JL, Grivich MI, Petrusca D. 54.  et al. 2006. J. Neurosci. 26:8254–66 [Google Scholar]
  55. Tang A, Jackson D, Hobbs J, Chen W, Smith JL. 55.  et al. 2008. J. Neurosci. 28:505–18 [Google Scholar]
  56. Shlens J, Field GD, Gaulthier JL, Greschner M, Sher A. 56.  et al. 2009. J. Neurosci. 29:5022–31 [Google Scholar]
  57. Ohiorhenuan IE, Mechler F, Purpura KP, Schmid AM, Hu Q, Victor JD. 57.  2010. Nature 466:617–21 [Google Scholar]
  58. Ganmor E, Segev R, Schniedman E. 58.  2011. PNAS 108:9679–84 [Google Scholar]
  59. Tkačik G, Marre O, Mora T, Amodei D, Berry MJ II, Bialek W. 59.  2013. J. Stat. Mech. 2013:P03011 [Google Scholar]
  60. Granot-Atedgi E, Tkačik G, Segev R, Schneidman E. 60.  2013. PLOS Comput. Biol. 9:e1002922 [Google Scholar]
  61. Tkačik G, Marre O, Amodei D, Schneidman E, Bialek W, Berry MJ II. 61.  2014. PLOS Comput. Biol. 10:e1003408 [Google Scholar]
  62. Mora T, Deny S, Marre O. 62.  2015. Phys. Rev. Lett. 114:078105 [Google Scholar]
  63. Tkačik G, Mora T, Marre O, Amodei D, Palmer SE. 63.  et al. 2015. PNAS 112:11508–13 [Google Scholar]
  64. Socolich M, Lockless SW, Russ WP, Lee H, Gardner KH, Ranganathan R. 64.  2005. Nature 437:512–18 [Google Scholar]
  65. Russ WP, Lowery DM, Mishra P, Yaffe MB, Ranganathan R. 65.  2005. Nature 437:579–83 [Google Scholar]
  66. Weigt M, White RA, Szurmant H, Hoch JA, Hwa T. 66.  2009. PNAS 106:67–72 [Google Scholar]
  67. Marks DS, Colwell LJ, Sheridan R, Hopf TA, Pagnani A. 67.  et al. 2011. PLOS ONE 6:e28766 [Google Scholar]
  68. Hopf TA, Colwell LJ, Sheridan R, Rost B, Sander C, Marks DS. 68.  2012. Cell 149:1607–162 [Google Scholar]
  69. Sulkowska JI, Morocos F, Weigt M, Hwa T, Onuchic JN. 69.  2012. PNAS 109:10340–45 [Google Scholar]
  70. Mora T, Walczak AM, Bialek W, Callan CG. 70.  2010. PNAS 107:5405–10 [Google Scholar]
  71. Ferguson AL, Mann JK, Omarjee S, Ndung'u T, Walker BD, Chakraborty AK. 71.  2013. Immunity 38:606–17 [Google Scholar]
  72. Bialek W, Cavagna A, Giardina I, Mora T, Silvestri E. 72.  et al. 2012. PNAS 109:4786–91 [Google Scholar]
  73. Bialek W, Cavagna A, Giardina I, Mora T, Pohl O. 73.  et al. 2014. PNAS 111:7212–17 [Google Scholar]
  74. Cavagna A, Giardina I. 74.  2014. Annu. Rev. Condens. Matter Phys. 5:183–207 [Google Scholar]
  75. Cavagna A, Del Castillo L, Dey S, Giardina I, Melillo S. 75.  et al. 2015. Phys. Rev. E 92:012705 [Google Scholar]
  76. Cavagna A, Cimarelli A, Giardina I, Parisi G, Santagati R. 76.  et al. 2010. PNAS 107:11865–70 [Google Scholar]
  77. Attanasi A, Cavagna A, Del Castello L, Giardina I, Melillo S. 77.  et al. 2014. Phys. Rev. Lett. 113:238102 [Google Scholar]
  78. Guttal V, Couzin ID. 78.  2010. PNAS 107:16172–77 [Google Scholar]
  79. Sharpee TO, Rust NC, Bialek W. 79.  2004. Neural Comput. 16:223–50 [Google Scholar]
  80. Sharpee TO, Sugihara H, Kurgansky AV, Rebrik SP, Stryker MP, Miller KD. 80.  2006. Nature 439:936–42 [Google Scholar]
  81. Fitzgerald JD, Rowekamp RJ, Sincich LC, Sharpee TO. 81.  2011. PLOS Comput. Biol. 7:e1002249 [Google Scholar]
  82. Rajan K, Bialek W. 82.  2013. PLOS ONE 8:e71959 [Google Scholar]
  83. Eickenberg M, Rowekamp RJ, Kouh M, Sharpee TO. 83.  2012. Neural Comput. 24:2384–421 [Google Scholar]
  84. Rajan K, Marre O, Tkačik G. 84.  2013. Neural Comput. 25:1661–92 [Google Scholar]
  85. Kinney JB, Tkačik G, Callan CG Jr. 85.  2007. PNAS 104:501–6 [Google Scholar]
  86. Kinney JB, Murugan A, Callan CG Jr, Cox EC. 86.  2010. PNAS 107:9158–63 [Google Scholar]
  87. Elemento O, Slonim N, Tavazoie S. 87.  2007. Mol. Cell 28:337–50 [Google Scholar]
  88. Crick FHC. 88.  1958. Symp. Soc. Exp. Biol. 12:138–63 [Google Scholar]
  89. Crick FHC. 89.  1963. Prog. Nucleic Acids Res. Mol. Biol. 1:163–217 [Google Scholar]
  90. Sanger F, Air GM, Barrell BG, Brown NL, Coulson AR. 90.  et al. 1977. Nature 265:687–95 [Google Scholar]
  91. Freeland SJ, Hurst LD. 91.  1998. J. Mol. Evol. 47:238–48 [Google Scholar]
  92. Freeland SJ, Knight RD, Landweber LF, Hurst LD. 92.  2000. Mol. Biol. Evol. 17:511–18 [Google Scholar]
  93. Tlusty T. 93.  2008. Phys. Rev. Lett. 100:048101 [Google Scholar]
  94. Hopfield JJ. 94.  1974. PNAS 71:4135–39 [Google Scholar]
  95. Ninio J. 95.  1975. Biochimie 57:587–95 [Google Scholar]
  96. Ehrenberg M, Kurland CG. 96.  1984. Q. Rev. Biophys. 17:45–82 [Google Scholar]
  97. MacKay D, McCulloch WS. 97.  1952. Bull. Math. Biophys. 14:127–35 [Google Scholar]
  98. Barlow HB. 98.  1959. Sensory mechanisms, the reduction of redundancy, and intelligence. Mechanisation of Thought Processes: Proceedings of a Symposium Held at the National Physical Laboratory on 24th, 25th, 26th and 27th November 1958, Volume 2 DV Blake, AM Uttley 537–74 London: HM Station. Off. [Google Scholar]
  99. Barlow HB. 99.  1961. Possible principles underlying the transformation of sensory messages. Sensory Communication W Rosenblith 217–34 Cambridge, MA: MIT Press [Google Scholar]
  100. Simoncelli EP, Olshausen BA. 100.  2001. Annu. Rev. Neurosci. 24:1193–216 [Google Scholar]
  101. Geisler WS. 101.  2008. Annu. Rev. Psychol. 59:167–92 [Google Scholar]
  102. Laughlin SB. 102.  1981. Z. Naturforsch. Teil C 36:910–12 [Google Scholar]
  103. Rieke F, Warland D, Bialek W. 103.  1993. Europhys. Lett. 22:151–56 [Google Scholar]
  104. Borst A, Theunissen FE. 104.  1999. Nat. Neurosci. 2:947–57 [Google Scholar]
  105. de Ruyter van Steveninck RR, Lewen GD, Strong SP, Koberle R, Bialek W. 105.  1997. Science 275:1805–8 [Google Scholar]
  106. Koch K, McLean J, Segev R, Freed MA, Berry MJ II. 106.  et al. 2006. Curr. Biol. 16:1428–34 [Google Scholar]
  107. Reinagel P, Reid RC. 107.  2000. J. Neurosci. 20:5392–400 [Google Scholar]
  108. Liu RC, Tzonev S, Rebrik S, Miller KD. 108.  2001. J. Neurophysiol. 86:2789–806 [Google Scholar]
  109. Rieke F, Bodnar DA, Bialek W. 109.  1995. Proc. R. Soc. Lond. Ser. B 262:259–65 [Google Scholar]
  110. Nemenman I, Lewen GD, Bialek W, de Ruyter van Steveninck RR. 110.  2008. PLOS Comput. Biol. 4:e1000025 [Google Scholar]
  111. Buračas GT, Zador AM, DeWeese MR, Albright TD. 111.  1998. Neuron 20:959–69 [Google Scholar]
  112. Kara P, Reinagel P, Reid RC. 112.  2000. Neuron 27:635–46 [Google Scholar]
  113. Ruderman DL, Bialek W. 113.  1994. Phys. Rev. Lett. 73:814–17 [Google Scholar]
  114. Smirnakis S, Berry MJ II, Warland DK, Bialek W, Meister M. 114.  1997. Nature 386:69–73 [Google Scholar]
  115. Gollisch T, Meister M. 115.  2010. Neuron 65:15–164 [Google Scholar]
  116. Wark B, Lundstrom BN, Fairhall AL. 116.  2007. Curr. Opin. Neurobiol. 17:423–29 [Google Scholar]
  117. Brenner N, Bialek W, de Ruyter van Steveninck RR. 117.  2000. Neuron 26:695–702 [Google Scholar]
  118. Fairhall AL, Lewen GD, Bialek W, de Ruyter van Steveninck RR. 118.  2001. Nature 412:787–92 [Google Scholar]
  119. Kvale MN, Schreiner CE. 119.  2004. J. Neurophysiol. 91:604–12 [Google Scholar]
  120. Dean I, Harper NS, McAlpine D. 120.  2005. Nat. Neurosci. 8:1684–89 [Google Scholar]
  121. Nagel KI, Doupe AJ. 121.  2006. Neuron 21:845–59 [Google Scholar]
  122. Wen B, Wang GI, Dean I, Delgutte B. 122.  2009. J. Neurosci. 29:13797–808 [Google Scholar]
  123. Rahmen JC, Keating P, Nodal FR, Schulz AL, King AJ. 123.  2010. Neuron 66:937–48 [Google Scholar]
  124. Rabinowitz NC, Willmore BDB, Schnup JWH, King AJ. 124.  2011. Neuron 70:1178–92 [Google Scholar]
  125. Maravall M, Petersen RS, Fairhall AL, Arabzadeh E, Diamond ME. 125.  2007. PLOS Biol. 5:e19 [Google Scholar]
  126. De Baene W, Premereur E, Vogels R. 126.  2007. J. Neurophysiol. 97:2900–16 [Google Scholar]
  127. Wark B, Fairhall AL, Rieke F. 127.  2009. Neuron 61:750–61 [Google Scholar]
  128. Atick JJ, Redlich AN. 128.  1990. Neural Comput. 2:308–20 [Google Scholar]
  129. Laughlin SB, de Ruyter van Steveninck RR. 129.  1996. J. Physiol. 494:P19 [Google Scholar]
  130. de Ruyter van Steveninck RR, Laughlin SB. 130.  1996. Nature 379:642–45 [Google Scholar]
  131. Chittka L, Menzel R. 131.  1992. J. Comp. Physiol. A 171:171–81 [Google Scholar]
  132. Garrigan P, Ratliff CP, Klein JM, Sterling P, Brainard DH, Balasubramanian V. 132.  2010. PLOS Comput. Biol. 6:e1000677 [Google Scholar]
  133. Ratliff CP, Borghuis BG, Kao Y-H, Sterling P, Balasubramanian V. 133.  2010. PNAS 107:17368–73 [Google Scholar]
  134. Borghuis BG, Ratliff CP, Smith RG, Sterling P, Balasubramanian V. 134.  2008. J. Neurosci. 28:3178–89 [Google Scholar]
  135. Liu YS, Stevens CF, Sharpee TO. 135.  2009. PNAS 106:16499–514 [Google Scholar]
  136. Iyengar G, Rao M. 136.  2014. PNAS 111:12402–7 [Google Scholar]
  137. Smith EC, Lewicki MS. 137.  2005. Neural Comput. 17:19–45 [Google Scholar]
  138. Smith EC, Lewicki MS. 138.  2006. Nature 439:978–82 [Google Scholar]
  139. van Hateren JH. 139.  1992. Biol. Cybern. 68:23–29 [Google Scholar]
  140. Olshausen BA, Field DJ. 140.  1996. Nature 381:607 [Google Scholar]
  141. Bell AJ, Sejnowski TJ. 141.  1997. Vis. Res. 23:3327–38 [Google Scholar]
  142. Linsker R. 142.  1989. Neural Comput. 1:402–11 [Google Scholar]
  143. Barlow H. 143.  2001. Network 12:241–53 [Google Scholar]
  144. Puchalla JL, Schneidman E, Harris RA, Berry MJ II. 144.  2005. Neuron 46:493–504 [Google Scholar]
  145. Tkačik G, Marre O, Amodei A, Schneidman E, Bialek W. 145.  et al. 2014. PLOS Comput. Biol. 10:e1003408 [Google Scholar]
  146. Karklin Y, Simoncelli E. 146.  2011. Efficient coding of natural images with a population of noisy linear-nonlinear neurons. Advances in Neural Information Processing Systems 24 J Shawe-Taylor, RS Zemel, PL Bartlett, F Pereira, KQ Weinberger 999–1007 Cambridge, MA: MIT Press [Google Scholar]
  147. Balasubramanian V, Kimber D, Berry MJ II. 147.  2001. Neural Comput. 13:799–815 [Google Scholar]
  148. Tkačik G, Prentice JS, Balasubramanian V, Schneidman E. 148.  2010. PNAS 107:14419–24 [Google Scholar]
  149. Osborne LC, Palmer SE, Lisberger SG, Bialek W. 149.  2008. J. Neurosci. 28:13522–31 [Google Scholar]
  150. Hafting T, Fyhn M, Molden S, Moser MB, Moser EI. 150.  2005. Nature 436:801–6 [Google Scholar]
  151. Fiete IR, Burak Y, Brookings T. 151.  2008. J. Neurosci. 28:6856–71 [Google Scholar]
  152. Sreenivasan S, Fiete IR. 152.  2011. Nat. Neurosci. 14:1330–37 [Google Scholar]
  153. Elowitz MB, Levine AJ, Siggia ED, Swain PD. 153.  2002. Science 297:1183–86 [Google Scholar]
  154. Ozbudak E, Thattai M, Kurtser I, Grossman AD, van Oudenaarden A. 154.  2002. Nat. Genet. 31:69–73 [Google Scholar]
  155. Blake WJ, Kaern M, Cantor CR, Collins JJ. 155.  2003. Nature 422:633–37 [Google Scholar]
  156. Tkačik G, Callan CG Jr, Bialek W. 156.  2008. Phys. Rev. E 78:011910 [Google Scholar]
  157. Tkačik G, Walczak AM. 157.  2011. J. Phys. Condens. Matter 23:153102 [Google Scholar]
  158. Gregor T, Tank DW, Wieschaus EF, Bialek W. 158.  2007. Cell 130:153–64 [Google Scholar]
  159. Tkačik G, Callan CG Jr, Bialek W. 159.  2008. PNAS 105:12265–70 [Google Scholar]
  160. Wolpert L. 160.  1969. J. Theor. Biol. 25:1–47 [Google Scholar]
  161. Dubuis JO, Tkačik G, Wieschaus EF, Gregor T, Bialek W. 161.  2013. PNAS 110:16301–8 [Google Scholar]
  162. Tkačik G, Dubuis JO, Petkova MD, Gregor T. 162.  2015. Genetics 199:39–59 [Google Scholar]
  163. François P, Siggia ED. 163.  2010. Development 137:2385–95 [Google Scholar]
  164. Tkačik G, Walczak AM, Bialek W. 164.  2009. Phys. Rev. E 80:031920 [Google Scholar]
  165. Walczak AM, Tkačik G, Bialek W. 165.  2010. Phys. Rev. E 81:041905 [Google Scholar]
  166. Tkačik G, Walczak AM, Bialek W. 166.  2012. Phys. Rev. E 85:041903 [Google Scholar]
  167. Sokolowski TR, Tkačik G. 167.  2015. Phys. Rev. E 91:062710 [Google Scholar]
  168. Ziv E, Nemenman I, Wiggins CH. 168.  2007. PLOS ONE 2e1007 [Google Scholar]
  169. Tostevin F, ten Wolde PR. 169.  2009. Phys. Rev. Lett. 102:21801 [Google Scholar]
  170. de Ronde WH, Tostevin F, ten Wolde PR. 170.  2010. Phys. Rev. E 82:031914 [Google Scholar]
  171. Bowsher CG, Swain PS. 171.  2014. Curr. Opin. Biotechnol. 28:149–55 [Google Scholar]
  172. Lestas I, Vinnicombe G, Paulsson J. 172.  2010. Nature 467:174–78 [Google Scholar]
  173. Kelly JL Jr. 173.  1956. Bell Syst. Tech. J. 35:917–26 [Google Scholar]
  174. Donaldson-Matasci MC, Bergstrom CT, Lachmann M. 174.  2010. Oikos 119:219–30 [Google Scholar]
  175. Kussell EL, Leibler S. 175.  2005. Science 309:2075–78 [Google Scholar]
  176. Rivoire O, Leibler S. 176.  2011. J. Stat. Phys. 142:1124–66 [Google Scholar]
  177. Vergassola M, Villermaux E, Shraiman BI. 177.  2007. Nature 445:406–9 [Google Scholar]
  178. Mafra-Neto A, Cardé RT. 178.  1994. Nature 369:142–44 [Google Scholar]
  179. Attneave F. 179.  1954. Psychol. Rev. 61:183–93 [Google Scholar]
  180. Shannon CE. 180.  1951. Bell Syst. Tech. J. 30:50–64 [Google Scholar]
  181. Chomsky N. 181.  1956. IRE Trans. Inf. Theory IT-2:113–24 [Google Scholar]
  182. Pereira F. 182.  2000. Philos. Trans. R. Soc. Lond. A 358:1239–53 [Google Scholar]
  183. Pereira F, Tishby N, Lee L. 183.  1993. Distributional clustering of English words. Proc. 31st Annu. Meet. Assoc. Comput. Linguist. LK Schubert 183–90 Stroudsburg, PA: Assoc. Comput. Linguist. [Google Scholar]
  184. Tishby N, Pereira FC, Bialek W. 184.  1999. The information bottleneck method. Proc. 37th Annu. Allerton Conf. Commun. Control Comput. B Hajek, RS Sreenivas 368–77 Champaign: Univ. Ill. Press [Google Scholar]
  185. Bialek W, de Ruyter van Steveninck RR, Tishby N. 185.  2007. Presented at 2006 IEEE International Symposium on Information Theory, Seattle, WA. arXiv:0712.4381 [q-bio.NC]
  186. Bialek W, Nemenman I, Tishby N. 186.  2001. Neural Comput. 13:2409–63 [Google Scholar]
  187. Bialek W, Nemenman I, Tishby N. 187.  2001. Physica A 302:89–99 [Google Scholar]
  188. Creutzig F, Globerson A, Tishby N. 188.  2009. Phys. Rev. E 79:041925 [Google Scholar]
  189. Palmer SE, Marre O, Berry MJ II, Bialek W. 189.  2015. PNAS 112:6908–13 [Google Scholar]
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