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REDES COMPLEJAS:
DEL CEREBRO A LAS REDES SOCIALES
JAVIER M. BULDÚ
UNIVERSIDAD REY JUAN CARLOS (MÓSTOLES)
CENTRO DETECNOLOG...
COMPLEJIMAD:
ASOCIACIÓN MADRILEÑA DE CIENCIAS
DE LA COMPLEJIDAD
SISTEMAS COMPLEJOS
Un sistema complejo está formado por partes interrelacionadas que, como conjunto, exhiben
propiedades y...
SISTEMAS COMPLEJOS
La sociedad, su organización y los procesos que en ella ocurren, se
pueden estudiar bajo la perspectiva...
REDES COMPLEJAS
Una Red Compleja es una red con una estructura no trivial, cuyos patrones de conexión ni
son regulares del...
REDES COMPLEJAS
Red de tráfico aéreo.
REDES COMPLEJAS
Red criminal de Messina (Italia). Emilio Ferrara, Indiana University
REDES COMPLEJAS
Una vez obtenida la red, la Ciencia de las Redes se encarga de analizarla basándose en cuatro pilares
fund...
REDES CEREBRALES
- Cross-correlation
- Wavelet coherence
- Sync. likelihood
- Generalized Sync.
- Phase Sync.
- Mutual Inf...
¿CÓMO SON LAS REDES
CEREBRALES? (SOCIALES)
1.- Suelen tener una estructura heterogénea y como consecuencia tienen
nodos mu...
1. SON REDES HETEROGÉNEAS
Las redes reales no son homogéneas, tienen “hubs”. Suelen seguir lo que se conoce
como una ley l...
Los hubs son omnipresentes en las redes sociales:
Facebook Data Science Section (2011).
50% tiene menos de 100 amigos
99% ...
Los hubs también aparecen en las redes cerebrales:
1. SON REDES HETEROGÉNEAS
❑ Dos actividades: música y finger tapping
❑ ...
Aparecen regiones altamente conectadas: “hubs”
1. SON REDES HETEROGÉNEAS
HUBS
Probabilidad de tener un número k de conexio...
• Las redes reales son redes de
“pequeño mundo” (small-
world).
• ¿Cómo de alejados estamos
unos de otros?
• Las redes soc...
• (1967) A un grupo de gente (296) de Omaha (Nebraska) y Wichita
(Kansas) se le pidió que enviara una carta a una persona ...
Veamos que ocurre en Facebook:
2. SON REDES DE PEQUEÑO MUNDO
Distancia media entre 1.600.000.000 usuarios de Facebook. Fue...
¿Ocurre lo mismo en las redes cerebrales?
Matriz de conexiones entre neuronas del C. Elegans.
(O. Sporns,The Networks of t...
¿Ocurre lo mismo en el cerebro humano?
2. SON REDES DE PEQUEÑO MUNDO
3. SON REDES CON ALTO CLUSTERING
El coeficiente de clustering mide la cantidad de contactos que, a su vez,
están en contact...
Se puede actuar localmente, mediante los vecinos de un nodo
(en “tripletes”), y aumentar la propagación a nivel global:
Ex...
Las redes cerebrales también tienen alto clustering:
Reconstrucción de redes anatómicas mediante resonancia magnética. 998...
4. SON REDES MODULARES: FORMAN GRUPOS
Es posible detectar grupos de nodos fuertemente conectados,
indicando la existencia ...
La formación de comunidades permite detectar el papel
que juegan los nodos en la estructura local/global de la red:
Red de...
Módulos estructurales en el córtex, obtenidos con resonancia magnética.
Se detectan 6 módulos (discos grises) junto con su...
Red funcional (reposo) obtenida mediante resonancia magnética funcional (fMRI). Se detectan 5 módulos principales:
central...
Asortatividad y Homofilia: me gustan los que son como yo…
Asortatividad: Los nodos más felices tienden a estar
conectados e...
Asortatividad y Topología: me conecto con nodos con
conectividad similar:
Ejemplo de un red de usuarios
de twitter (40M tw...
La asortatividad surge de manera espontánea, no es
necesario forzarla:
2014
5. SON REDES ASORTATIVAS: FORMAN RICH-CLUBS
Lo mismo ocurre en las redes cerebrales:
Las zonas más conectadas, tienden a estar
más conectadas entre ellas. (finger tapp...
Como consecuencia de la asortatividad y la modularidad,
aparecen rich-clubs:
Red de conexiones cortico-corticales. (A) Apa...
– Groucho Marx
“Todo esto es tan sencillo que hasta un niño de 5
años lo entendería… Que me traigan a un niño de 5
años!”
...
El proceso entero es un campo de minas!
EL PROCESO DE OBTENCIÓN DE LAS REDES
PRESENTA MUCHAS DIFICULTADES
2.4 The Brain as...
PROBLEMA: COMPARAR LAS REDES ENTRE ELLAS
Datos: Red anatómica (Hagmann et aI., 2008) y red funcional (Honey et aI., 2009) ...
PROBLEMA: COMPARAR LAS REDES ENTRE ELLAS
EJEMPLO SOCIAL:
Facebook: Cuatro vistas diferentes
de una misma red de Facebook.
...
Resonancia magnética funcional en (A) reposo y (B) durante una tarea de memoria.
Relaciones funcionales entre las zonas má...
Red funcional (fMRI) con diferentes grupos de edad. Los nodos se agrupan siguiendo un algoritmo
basado en muelles. La zona...
La topología de la red condiciona la dinámica, pero también a la inversa. Por ejemplo,
el aprendizaje hebbiano refuerza la...
Autorretratos de William Utermohlen (pintor estadounidense (1993-2007)). En
1995 (con 62 años) empieza a ser atendido por ...
RESUMIENDO… (Y LO DEJO!)
I. LA CIENCIA DE LAS REDES PUEDE AYUDARNOS A
COMPRENDER
MEJOR EL CEREBRO… O A INTENTARLO!
II. LA ...
Beware of the small-world, neuroscientist!
David Papo1,*
, Massimiliano Zanin2,3
, Johann H. Martínez4,5
, and Javier M. B...
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Redes complejas: del cerebro a las redes sociales

En esta presentación Javier Buldú aplica los cinco principios a las redes neuronales y redes sociales

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Redes complejas: del cerebro a las redes sociales

  1. 1. REDES COMPLEJAS: DEL CEREBRO A LAS REDES SOCIALES JAVIER M. BULDÚ UNIVERSIDAD REY JUAN CARLOS (MÓSTOLES) CENTRO DETECNOLOGÍA BIOMÉDICA (POZUELO) COMPLEJIMAD (MADRID) CLUB SICOMORO, 25/10/2017
  2. 2. COMPLEJIMAD: ASOCIACIÓN MADRILEÑA DE CIENCIAS DE LA COMPLEJIDAD
  3. 3. SISTEMAS COMPLEJOS Un sistema complejo está formado por partes interrelacionadas que, como conjunto, exhiben propiedades y comportamientos no evidentes a partir de la suma de las partes individuales. Una neurona Un cerebro
  4. 4. SISTEMAS COMPLEJOS La sociedad, su organización y los procesos que en ella ocurren, se pueden estudiar bajo la perspectiva de los sistemas complejos: Red deTwitter. Los nodos son usuarios conectados por tweets Subred deTwitter: 415.808 conexiones y 283.317 nodos.
  5. 5. REDES COMPLEJAS Una Red Compleja es una red con una estructura no trivial, cuyos patrones de conexión ni son regulares del todo ni completamente aleatorios. Su estructura es fundamental para entender los procesos que en ella ocurren: N=9 L=9 1 3 2 4 5 8 6 9 UNA RED 34 356 460 175 7 1203 516 690 726 139 10 697 692 65 511 1404 536 546 570 1518 169 812 1023 1655 1337 910 329 328450547 1610 52 1490554 105 676 331 434 7231590 6 359 149 353 251 386 387 357 240 185 1609 1168 724 509 1051900 828 928 873 147 368 365 252 393 1681 484 44 388 19 347576 111 56 628 1381 567 553 689 679 521 805 533 47 298 743 498 772 28 272 881 309 390 36 33 1025 797 548 414 258 532 392502 391410 223 1372 78 874 89 144 73 71 1004 226 257 219 581 710 209 513 1276 1701 704 216 1034 238 358 256 1374 243 16891348 14 1112 822 1670 15071232 425 75 327 94 416 10921003 921 1591 25 1027 834 1487 441 644 702 688 1228 1444 163 525 1495 1223 1197 31 637 1498 343 342 128 67 1015 137 93 522 212 3954 380 378 27 265 24 1174 218 418 304 722490253 79 95 563 497 374369 377 504 228 38 334 53 363 560 354215 1234 88351 70 1631 1059 45 247 130 499 20558 193 148 604 254 582577 242 506 1017196 1028 1111 214 136 308 1363 208 372 180 599 431 4 150 96 578 980 1094 1387559 181 1515 293 534 1287 903 552 1282 49 178 1544 771 807 437 157 382 282 106 1210 1488 1671 194 313 389 1514 1569 786 1482 339 1221 832 591 937 917 853 1296 1127 699 1022 846 292 400 1319 287 167 286 800 81 5391297 9591231 1045931 74 500 1225 91 752143211751 440 98 891 669 1457 1133 299 1674 1511 470 799 661 1046444 681 119 302517 115 1418 151 1550 915 659 801 326 462 1186 1099 936 783 396 415 1580 6091339 397 887 1250 239 235 615 244 241 575 986 620 586 1357 245 306 1451504 84 1080 92 894 398 1513 544 1395 276 879755 8881 107 925 1121 232 233 1455 234 1603 566 1347 1662 951 589621 318 1628 596 580 623 1105949 618 632 1524231 1643 728 432 1119 569 1512 1103 317 1283 1040 143 795 950 1009 344 29 836 15 104 273 16 1477 277 395188 1526 11 283459 32 62 493 428394 13 13701116 179 255 160 116 50 158 587 1441 991 133 80 352 198 17 579 600 340 159 134 69 126 1440 125 165 237 164 1650 1435 1213 830 821 957 494 655 767 404 972 1331 598 745 1658 633 346 1383 1465 861 146 573 487 430 1005 645 902 1138 1708 888 1139 407 87 1642 267 973 648 1254 1346 271 376 1354 485 20 491 409 285 1604 68 446 383 311 355 455 281 100 508 221 141 337 66 201 997 1533 220 1318 140 263 320 310284 72 335 191 305 1377 189 1608 905 809 1257 206 123 457 8131574 375 1271 162 875 943 1697 1151 315 40 316 289 294 403 268 564 1237 399 118 103 99 57 523 571 291 1334 406 574 565 225 1060 59 1379 295 665 1576 501 127496 176 22 110 866 270 280 156 124 572 186 23 585 1541 349 514 381 236 222 510 478 540 248 325 1554 154 345 324 8190 588 1128 300 519 122 590 330 1102 468 297 55 412 2 132 210 1031 113 303 472 120 466 296 458 43 187 748 421 41 213 597 1429 995 1164 109 952 520 1012 422 429 442 419336 170 529 288 161 275 469 155 593 789 102 211 142 402 524 379 1155 1220 1217 1194 1261 448 606 436 1684 867 1158 1551 11541134 512 1039 862 1170 568 1534 427 819 756 967 350 1233 518 480 1664 461 1582 1241 750 557 675 1201 301 1698 531 443 526 18 121 1079 640 420 456 595 1340 307 5281054 1107 467 338 114 341 613 465 463 5621089 872 8241087 264 639 765 962 4131475 224 373 6161303 1647 384 974 10421317 1369 538 1649 541 1083 1646 884 229 737 1350 197 1227 922 1314 129 842 709 594 192 260 537 1434 934 1157 766 204 184 1492 663 1058 1496 202 906 1295 1288 475 741 774 744 83 3 366 1204 1577 674 360445 848 1694 770 112 759 16021497 1274 1212 1620 701 1399 885 1486 481 449 321 643 333 138 1450 1311 1616 153 1439 1026 433 977 82 172 171 1695 1605 489 1413 1202 435 438 1001 1216 1528 641 454 1433 452 1192 829 1293 1073 1630 1414 168507 1178 1593 876 5 492 362 1219 1207 473 101 730 1537 864 1325 1184 1562 503 42330 1406 495 527 1437 15301300 530 1333 1321 1273 269 551 550 691 634 555451 556 549 776 483 1007 1343 477 173 944 1365 1494 21 930 1682 97 1328 1391 1588 1394 401 1122 515 195 1049 890 12 653 1565 1086 610 90 207 1159 230 199 86 364 542 227 1214 882 261 1547 583 981 1291 826 259 249 1548 246 869278 1596 16251426 10211251 1267 1425 135 592 1419 279 1654 920 707 982 2031675 1286 200 1090 453 361 323 584 1063 987 408 1332 1038 946 447 1402 166 108 9 683 385 46 1384 367 1570 26 48 177 77 6360 1057 64 42 877 482 1289 841 UNA RED COMPLEJA “Las redes complejas son como el porno, no tengo una definición precisa, pero lo reconozco cuando lo veo” M.A. Porter (University of Oxford)
  6. 6. REDES COMPLEJAS Red de tráfico aéreo.
  7. 7. REDES COMPLEJAS Red criminal de Messina (Italia). Emilio Ferrara, Indiana University
  8. 8. REDES COMPLEJAS Una vez obtenida la red, la Ciencia de las Redes se encarga de analizarla basándose en cuatro pilares fundamentales: la teoría de grafos, la física estadística, la dinámica no lineal y el Big Data. Puedo analizar la estructura de una red, independientemente de su naturaleza.
  9. 9. REDES CEREBRALES - Cross-correlation - Wavelet coherence - Sync. likelihood - Generalized Sync. - Phase Sync. - Mutual Info. - Granger Causality - EEG - MEG - fMRI - Histological Analysis - DTI (MRI) REDES ANATÓMICAS REDES FUNCIONALES From Bullmore & Sporns, Nature Rev. 10, 186 (2009)
  10. 10. ¿CÓMO SON LAS REDES CEREBRALES? (SOCIALES) 1.- Suelen tener una estructura heterogénea y como consecuencia tienen nodos muy conectados (hubs). 2.- Las redes cerebrales son redes de pequeño mundo (small-world). 3.- El coeficiente de clustering suele ser muy alto. 4.- Son redes con alta modularidad: forman comunidades o grupos. 5.- Suelen ser redes asortativas (es decir, los nodos muy conectados suelen estar conectados entre ellos).
  11. 11. 1. SON REDES HETEROGÉNEAS Las redes reales no son homogéneas, tienen “hubs”. Suelen seguir lo que se conoce como una ley libre de escala, ya que la distribución de contactos es muy heterogénea: Red de contactos sexuales.: Parejas durante toda la vida. Muestra: 4781 suecos. Liljeros, Nature, 411, 907 (2001). totalacumulado HUBS numero de parejas mujeres hombres
  12. 12. Los hubs son omnipresentes en las redes sociales: Facebook Data Science Section (2011). 50% tiene menos de 100 amigos 99% tiene menos de 1500 1. SON REDES HETEROGÉNEAS HUBS (1% tiene más de 1500)
  13. 13. Los hubs también aparecen en las redes cerebrales: 1. SON REDES HETEROGÉNEAS ❑ Dos actividades: música y finger tapping ❑ fMRI (resonancia magnética funcional) ❑ 36 x 64 x 64 regiones (147456 voxels) ❑ Se mide la correlación entre regiones: ❑ Se analiza la matriz de conexiones. Music Finger tapping
  14. 14. Aparecen regiones altamente conectadas: “hubs” 1. SON REDES HETEROGÉNEAS HUBS Probabilidad de tener un número k de conexiones (Chialvo et al., PRL 2005)
  15. 15. • Las redes reales son redes de “pequeño mundo” (small- world). • ¿Cómo de alejados estamos unos de otros? • Las redes sociales están altamente conectadas y es fácil llegar a cualquier persona mediante la red de contactos en un bajo número de pasos. Stanley Milgram (NY, 1933-1984) fue un sorprendente psicólogo americano que destacó, sobre todo, por sus trabajos acerca de la obediencia a la autoridad. 2. SON REDES DE PEQUEÑO MUNDO
  16. 16. • (1967) A un grupo de gente (296) de Omaha (Nebraska) y Wichita (Kansas) se le pidió que enviara una carta a una persona desconocida de Boston (Massachussetts). • Regla básica del experimento: La persona debía reenviar la carta a otra persona de su entorno que considerara más cercana a la persona objetivo, y así sucesivamente • Hipótesis: Las redes sociales están altamente conectadas y es fácil llegar a cualquier persona mediante la red de contactos en un bajo número de pasos. 1 2 • 232 de 296 carta nunca llegaron a su destino. • 64 cartas llegaron a su destino (con caminos de entre 2 y 10 pasos). • El número promedio de pasos fue … 5.2 !!! EXPERIMENTO RESULTADOS SMALL-WORLD 34 356 460 175 7 1203 516 690 726 139 10 697 692 65 511 1404 536 546 570 1518 169 812 1023 1655 1337 910 329 328450547 1610 52 1490554 105 676 331 434 7231590 6 359 149 353 251 386 387 357 240 185 1609 1168 724 509 1051900 828 928 873 147 368 365 252 393 1681 484 44 388 19 347576 111 56 6281 381 567 553 689 679 521 805 533 47 298 743 498 772 28 272 881 309 390 36 33 1025 797 548 414 258 532 392502 391410 223 1372 78 874 89 144 73 71 1004 226 257 219 581 710 209 513 1276 1701 704 216 1034 238 358 256 1374 243 16891348 14 1112 822 1670 15071232 425 75 327 94 416 10921003 921 1591 25 1027 834 1487 441 644 702 688 1228 1444 163 525 1495 1223 1197 31 637 1498 343 342 128 67 1015 137 93 522 212 3954 380 378 27 265 24 1174 218 418 304 722490253 79 95 563 497 374369 377 504 228 38 334 53 363 560 354215 1234 88351 70 1631 1059 45 247 130 499 20558 193 148 604 254 582577 242 506 1017196 1028 1111 214 136 308 1363 208 372 180 599 431 4 150 96 578 980 1094 1387559 181 1515 293 534 1287 903 552 1282 49 178 1544 771 807 437 157 382 282 106 1210 1488 1671 194 313 389 1514 1569 786 1482 339 1221 832 591 937 917 853 1296 1127 699 1022 846 292 400 1319 287 167 286 800 81 5391297 9591231 1045931 74 500 1225 91 752143211751 440 98 891 669 1457 1133 299 1674 1511 470 799 661 1046444 681 119 302517 115 1418 151 1550 915 659 801 326 462 1186 1099 936 783 396 415 1580 6091339 397 887 1250 239 235 615 244 241 575 986 620 586 1357 245 306 1451504 84 1080 92 894 398 1513 544 1395 276 879755 8881 107 925 1121 232 233 1455 234 1603 566 1347 1662 951 5896213 18 1628 596 580 623 1105949 618 632 1524231 1643 728 432 1119 569 1512 1103 317 1283 1040 143 795 950 1009 344 29 836 15 104 273 16 1477 277 395188 1526 11 283459 32 62 493 428394 13 13701116 179 255 160 116 50 158 587 1441 991 133 80 352 198 17 579 600 340 159 134 69 126 1440 125 165 237 164 1650 1435 1213 830 821 957 494 655 767 404 972 1331 598 745 1658 633 346 1383 1465 861 146 573 487 430 1005 645 902 1138 1708 888 1139 407 87 1642 267 973 648 1254 1346 271 376 1354 485 20 491 409 285 1604 68 446 383 311 355 455 281 100 508 221 141 337 66 201 997 1533 220 1318 140 263 320 310284 72 335 191 305 1377 189 1608 905 809 1257 206 123 457 8131574 375 1271 162 875 943 1697 1151 315 40 316 289 294 403 268 564 1237 399 118 103 99 57 523 571 291 1334 406 574 565 225 1060 59 1379 295 665 1576 501 127496 176 22 110 866 270 280 156 124 572 186 23 585 1541 349 514 381 236 222 510 478 540 248 325 1554 154 345 324 8190 588 1128 300 519 122 590 330 1102 468 297 55 412 2 132 210 1031 113 303 472 120 466 296 458 43 187 748 421 41 213 597 1429 995 1164 109 952 520 1012 422 429 442 419336 170 529 288 161 275 469 155 593 789 102 211 142 402 524 379 7321438 998 1096 131 1665 174 1614 85 1011 1075 714 1238 76 474 1371 1155 1220 1217 1194 1261 448 606 436 1684 867 1158 1551 11541134 908649 1308 703 505 15361048 486 37 476 961 512 1039 862 1170 568 1534 427 819 756 967 350 1233 518 480 1664 461 1582 1241 750 557 675 1201 301 1698 531 443 526 18 121 1079 640 420 456 595 1340 307 5281054 1107 467 338 114 341 613 465 463 5621089 872 8241087 264 639 765 962 4131475 224 373 6161303 1647 384 974 10421317 1369 538 1649 541 1083 1646 884 229 737 1350 197 1227 922 1314 129 842 709 594 192 260 537 1434 934 1157 766 204 184 1492 663 1058 1496 202 906 1295 1288 475 741 774 744 83 3 366 1204 1577 674 360445 848 1694 770 112 759 16021497 1274 1212 1620 701 1399 885 1486 481 449 768698 860 314 1707 684 321 643 333 138 1450 1311 1616 153 1439 1026 433 977 82 172 171 1695 1605 489 1413 1202 435 1147 438 1001 405 1216 1528 1393 479 1523 918 763 1018 641 1245 454 1433 452 1192 829 1293 1463 798 322332 1571 1639 736 1403 1073 1630 1414 168507 1178 1593 876 5 492 362 1219 1207 473 101 730 1537 864 1325 1184 1562 503 42330 1406 495 527 1437 15301300 530 1333 1321 1273 269 551 550 691 634 555451 556 549 776 483 1007 1343 477 173 1098 855 1660 411 944 1365 1494 21 930 1682 97 1328 1391 1588 1394 401 1122 515 195 1049 890 12 653 1565 1086 610 90 207 1159 230 199 86 364 542 227 1214 882 261 1547 583 981 1291 826 259 249 1548 246 869278 1596 16251426 10211251 1267 1425 135 592 1419 279 1654 920 707 982 2031675 1286 200 1090 453 361 323 584 1063 987 408 1332 1038 946 447 1402 166 108 9 683 385 46 1384 367 1570 26 48 177 77 6360 1327 1566 312 1242 1141 708 152 1389 120615601306 656 1696 61764 1057 64 42 877 482 1289 841 1648 1190 348 1464 274250 1640 2. SON REDES DE PEQUEÑO MUNDO
  17. 17. Veamos que ocurre en Facebook: 2. SON REDES DE PEQUEÑO MUNDO Distancia media entre 1.600.000.000 usuarios de Facebook. Fuente: Lars Backstrom, Facebook Data Science.
  18. 18. ¿Ocurre lo mismo en las redes cerebrales? Matriz de conexiones entre neuronas del C. Elegans. (O. Sporns,The Networks of the Brain) • C. Elegans, un nematodo del que sabemos mucho. • A l r e d e d o r d e 3 0 0 neuronas. • Te n e m o s t o d a s l a s conexiones entre neuronas: podemos estudiar su red. L=2.65 (Lran=2.25) 2. SON REDES DE PEQUEÑO MUNDO
  19. 19. ¿Ocurre lo mismo en el cerebro humano? 2. SON REDES DE PEQUEÑO MUNDO
  20. 20. 3. SON REDES CON ALTO CLUSTERING El coeficiente de clustering mide la cantidad de contactos que, a su vez, están en contacto entre ellos: los amigos de mis amigos son mis amigos: Coeficiente de clustering en tres casos sencillos. 1 2 3 4 1 2 3 4 1 2 3 4 C1,2,3,4 = {0,0,0,0} C=0 C1,2,3,4 = {1,1,1,1} C=1 C1,2,3,4 = {1,0,1,1/3} C=7/12
  21. 21. Se puede actuar localmente, mediante los vecinos de un nodo (en “tripletes”), y aumentar la propagación a nivel global: Experimento online: un grupo de personas (1528) cuyos contactos son controlados artificialmente, deciden darse de alta en diferentes webs. Centola 329, 3 (2010). EXPERIMENTO ONLINE RESULTADOS 3. SON REDES CON ALTO CLUSTERING
  22. 22. Las redes cerebrales también tienen alto clustering: Reconstrucción de redes anatómicas mediante resonancia magnética. 998 regiones de interés (ROI) (Difussion Spectrum Imaging). Hagmann et al. (2008) PLoS Biol. 6, e159 Alto número de triángulos, comparado con redes aleatorias. 3. SON REDES CON ALTO CLUSTERING
  23. 23. 4. SON REDES MODULARES: FORMAN GRUPOS Es posible detectar grupos de nodos fuertemente conectados, indicando la existencia de patrones particulares dentro de la red: Las redes reales están organizadas en comunidades, aunque en muchas ocasiones es difícil detectarlas. Mejora la clasificación de hubs Hubs locales Hubs globales Participación ImportanciaLocal “P.Amos”
  24. 24. La formación de comunidades permite detectar el papel que juegan los nodos en la estructura local/global de la red: Red de colaboración en música Teitelbaum et al., Chaos, 18, 043105 (2008). 4. SON REDES MODULARES: FORMAN GRUPOS
  25. 25. Módulos estructurales en el córtex, obtenidos con resonancia magnética. Se detectan 6 módulos (discos grises) junto con sus hubs conectores y locales. Hagmann et al., PLoS Biol 6, 159 (2008). 4. SON REDES MODULARES: FORMAN GRUPOS
  26. 26. Red funcional (reposo) obtenida mediante resonancia magnética funcional (fMRI). Se detectan 5 módulos principales: central, parieto-frontal, medial occipital, lateral occipital y fronto-temporal. Meunier et al., Front. Neuroinformatics 3:37 (2009). Modularity of brain networks B processes of modularization might be disrupted in the pathogenesis of neuropsychiatric disor- ders such as autism or schizophrenia, supporting abnormal modularity of brain network organiza- tion as a diagnostic biomarker. In support of this expectation, some evidence for dysmodularity, or abnormal modular organization, has already Central module Parieto−frontal module Lateral occipital module A C B FIGURE 4 | Hierarchical modularity of a human brain functional network. (A) Cortical surface mapping of the community structure of the network at the highest level of modularity; (B) anatomical representation of the connectivity between nodes in color-coded modules.The brain is viewed from the left side with the frontal cortex on the left of the panel and occipital cortex on the right. Intra-modular edges are drawn in black; ( (shown centrally) illu no major sub-modul sub-modules. Repro impor ity of to co exam phren some, processes of modularization might be disrupted in the pathogenesis of neuropsychiatric disor- ders such as autism or schizophrenia,supporting abnormal modularity of brain network organiza- tion as a diagnostic biomarker. In support of this expectation, some evidence for dysmodularity, Central module Parieto−frontal module Lateral occipital module A C B FIGURE 4 | Hierarchical modularity of a human brain functional network. (A) Cortical surface mapping of the community structure of the network at the highest level of modularity; (B) anatomical representation of the connectivity between nodes in color-coded modules.The brain is viewed from the left side with the frontal cortex on the left of the panel and occipital cortex on the right. Intra-modular edges are colo are drawn in black; (C) sub-m (shown centrally) illustrates, no major sub-modules where sub-modules. Reproduced w Meunier et al. Modularity of brain networks Central module Medial occipital moduleParieto−frontal module Fronto−temporal moduleLateral occipital module A C B FIGURE 4 | Hierarchical modularity of a human brain functional network. (A) Cortical surface mapping of the community structure of the network at the highest level of modularity; (B) anatomical representation of the connectivity between nodes in color-coded modules.The brain is viewed from the left side Intra-modular edges are colored differently for each module; inter-modular edges are drawn in black; (C) sub-modular decomposition of the five largest modules (shown centrally) illustrates, for example, that the medial occipital module has no major sub-modules whereas the fronto-temporal module has many processes of modu in the pathogenes ders such as autism abnormal modular tion as a diagnostic expectation, some or abnormal mod Central module Lateral occipital module A C FIGURE 4 | Hierarchical modularity of a human brain functio (A) Cortical surface mapping of the community structure of the n highest level of modularity; (B) anatomical representation of the between nodes in color-coded modules.The brain is viewed from with the frontal cortex on the left of the panel and occipital cortex Meunier et al. Modularity of brain netwo Central module Medial occipital moduleParieto−frontal module Fronto−temporal moduleLateral occipital module A C B FIGURE 4 | Hierarchical modularity of a human brain functional network. (A) Cortical surface mapping of the community structure of the network at the highest level of modularity; (B) anatomical representation of the connectivity between nodes in color-coded modules.The brain is viewed from the left side Intra-modular edges are colored differently for each module; inter-modular edge are drawn in black; (C) sub-modular decomposition of the five largest modules (shown centrally) illustrates, for example, that the medial occipital module has no major sub-modules whereas the fronto-temporal module has many No importa que la red sea anatómica o funcional, los módulos aparecen en ambos casos: 4. SON REDES MODULARES: FORMAN GRUPOS
  27. 27. Asortatividad y Homofilia: me gustan los que son como yo… Asortatividad: Los nodos más felices tienden a estar conectados entre ellos… y viceversa. C.A. Bliss, I. M. Kloumann, K. D. Harris, C. M. Danforth, P. S. Dodds.  Twitter Reciprocal Reply Networks Exhibit Assortativity with Respect to Happiness. Journal of Computational Science. 2012. because of the uni-modal distribution of havg for the labMT words. Thus a moderate value for h is chosen ( h is set to 1 for this study). squares ( havg = 0) and green diamonds ( havg = 1). The average and standard deviation of the Spearman correlation coefficient calculated for the 100 randomized happiness scores (null model) are shown as red circles with error bars (the error bars are smaller than the symbol). This data supports the hypothesis that happiness is less assortative as network distance increases. Lastly, we explore whether these correlations are due to simi- larity of word usage. For this analysis, we compute the similarity of word bags for users connected in the reciprocal reply networks. We compare the distribution of observed similarity scores to similarity grate results in dead links w This problem of unfriendi impact conclusions drawn infer contagion. Our characterization o several trends over the 25 February 2009. The num work increased as time pr Twitter’s enormous grow Similarly, with an increa smaller proportion of close decrease). This may be du to an increasing N, with (i.e., friends of friends) ca in the giant component r 0 0.1 0.2 0.3 0.4 0.5 r s r s 1 2 3 0 0.1 0.2 0.3 0.4 0.5 Week 1 Week 2 Week 3 Week 4 Week 5 Week 6 Week 7 Week 8 Week 9 Week 10 Week 11 Week 12 Week 13 Week 14 Week 15 Week 16 Week 17 Week 18 Week 19 Week 20 Week 21 Week 22 Week 23 Week 24 Week 25 Links away (a) ∆h = 1,α = 1 Fig. 10. Happiness assortativity as measured by Spearman’s correlation coefficients is shown for week networks, with by users set to ˛ = 1 and (b) ˛ = 50. The dashed lines indicate weakening happiness–happiness correlations as the path len for each week in the data set. 5. SON REDES ASORTATIVAS: FORMAN RICH-CLUBS Twitter
  28. 28. Asortatividad y Topología: me conecto con nodos con conectividad similar: Ejemplo de un red de usuarios de twitter (40M tweets) C.A. Bliss, I. M. Kloumann, K. D. Harris, C. M. Danforth, P. S. Dodds.  Twitter Reciprocal Reply Networks Exhibit Assortativity with Respect to Happiness. Journal of Computational Science. 2012. 5. SON REDES ASORTATIVAS: FORMAN RICH-CLUBS
  29. 29. La asortatividad surge de manera espontánea, no es necesario forzarla: 2014 5. SON REDES ASORTATIVAS: FORMAN RICH-CLUBS
  30. 30. Lo mismo ocurre en las redes cerebrales: Las zonas más conectadas, tienden a estar más conectadas entre ellas. (finger tapping) música tapping 5. SON REDES ASORTATIVAS: FORMAN RICH-CLUBS
  31. 31. Como consecuencia de la asortatividad y la modularidad, aparecen rich-clubs: Red de conexiones cortico-corticales. (A) Aparecen módulos (segregación) conectados entre ellos por hubs conectores (integración). (B) Módulos: visual (amarillo), auditivo (rojo), somatosensorial-motor (verde), y frontolímbico (azul) areas en el córtex del gato. (C) Los hubs integran toda la información formando un rich-club solo detectable con el análisis de redes. Zamora et al, Front. Hum. Neurosci. 5, 83 (2011). Zamora-López et al. Anatomical brain connectivity FIGURE 2 | Segregation and integration of multisensory information. (A) Cortico-cortical networks are organized into modules composed of areas devoted to the processing of information of one modality.This modular organization permits the brain to handle information of different modalities in parallel, at the same time by different regions. (B) At the cortical surface modaly related areas are found close to each other, as illustrated by the distribution of visual (yellow), auditory (red), somatosensory-motor (green), and frontolimbic (blue) areas in the cortex of cats. (C) Cortical hubs form a central module at the top of the cortical hierarchy, which is capable of integrating multisensory information as the coordinated activity of the hubs. (D)This module can only be detected by connectivity analysis because cortical hubs are dispersed throughout the cortical surface. 5. SON REDES ASORTATIVAS: FORMAN RICH-CLUBS
  32. 32. – Groucho Marx “Todo esto es tan sencillo que hasta un niño de 5 años lo entendería… Que me traigan a un niño de 5 años!” DEL CEREBRO A LA RED CEREBRAL
  33. 33. El proceso entero es un campo de minas! EL PROCESO DE OBTENCIÓN DE LAS REDES PRESENTA MUCHAS DIFICULTADES 2.4 The Brain as a Complex Network 39 0MROW *MPXIVMRK1IXVMGW 7XEXMWXMGW (ITIRHIRGMIW2SHIW Brain activity Recorded signals Connectivity Matrix Graphs Topological properties Neuromarkers Healthy vs. Diseased Rest vs. Task Figure 2.5: The general framework of brain networks. Clockwise guideline. Nodes can be regarded as sensor or electrodes recording the electromagnetic signals of the brain, which may contain dependencies based on correlation or causality. These interdependencies, or link weights, lead to a weighted connectivity matrix, which is the mathematical representation of a network. This network is usually filtered using statistical thresholds to work only with the relevant links. Network
  34. 34. PROBLEMA: COMPARAR LAS REDES ENTRE ELLAS Datos: Red anatómica (Hagmann et aI., 2008) y red funcional (Honey et aI., 2009) para el mismo grupo de individuos. 998 regiones de interés (ROIs). La matriz estructural es solo positiva, mientras que la funcional puede ser positiva/negativa. RH: hemisferio izquierdo, LH: hemisferio derecho. Red anatómica (DTI) Red funcional (fMRI)
  35. 35. PROBLEMA: COMPARAR LAS REDES ENTRE ELLAS EJEMPLO SOCIAL: Facebook: Cuatro vistas diferentes de una misma red de Facebook. Respectivamente: red de amigos, red de relaciones (visitas de páginas), comunicación unidireccional y comunicación bidireccional. Misma red, con distintos niveles de información. D. Easley & J. Kleinberg, Networks, crowds and markets.
  36. 36. Resonancia magnética funcional en (A) reposo y (B) durante una tarea de memoria. Relaciones funcionales entre las zonas más activas de la red para ambos casos. Nodos: rMTL, right medial temporal lobe; IMTL, left medial temporal lobe; dmPFC, dorsomedial prefrontal cortex; vmPFC, ventro medial prefrontal cortex; rTC, right temporal cortex; lTC, left temporal cortex; rIPL, right inferior parietal lobe; lIPL, left inferior parietal lobe. Fransson et al., Neuroimage (2008). PROBLEMA: LAS REDES FUNCIONALES CAMBIAN CONTINUAMENTE Las redes funcionales cambian en función de la tarea que se esté realizando:
  37. 37. Red funcional (fMRI) con diferentes grupos de edad. Los nodos se agrupan siguiendo un algoritmo basado en muelles. La zona azul representa la region frontal, la cual se segrega funcionalmente con la edad. Fair et al. PLoS Comp. Bio.(2009). PROBLEMA: LAS REDES FUNCIONALES CAMBIAN CON LA EDAD Con el paso del tiempo, las redes funcionales también modifican su estructura:
  38. 38. La topología de la red condiciona la dinámica, pero también a la inversa. Por ejemplo, el aprendizaje hebbiano refuerza las conexiones entre nodos que se coordinan habitualmente. Sporns, The networks of the Brain. Las redes no evolucionan…. co-evolucionan! PROBLEMA:TOPOLOGÍAY DINÁMICA ESTÁN RELACIONADAS determina afecta evolución topológica afecta dinámica neuronal topología estado determina
  39. 39. Autorretratos de William Utermohlen (pintor estadounidense (1993-2007)). En 1995 (con 62 años) empieza a ser atendido por problemas de memoria y escritura. PROBLEMA: LAS REDES FUNCIONALES SE DEGENERAN C.J. Stam et al., Cereb. Cortex (2006)
  40. 40. RESUMIENDO… (Y LO DEJO!) I. LA CIENCIA DE LAS REDES PUEDE AYUDARNOS A COMPRENDER MEJOR EL CEREBRO… O A INTENTARLO! II. LA MAYOR PARTE DE LAS REDES REALES COMPARTEN CIERTAS PROPIEDADES EMERGENTES
  41. 41. Beware of the small-world, neuroscientist! David Papo1,* , Massimiliano Zanin2,3 , Johann H. Martínez4,5 , and Javier M. Buldú1,6 1 Laboratory of Biological Networks, Center for Biomedical Technology & GISC, UPM, Madrid, Spain 2 Faculdade de Ciencias e Tecnologia, Departamento de Engenharia Electrotecnica, Universidade Nova de Lisboa, Lisboa, Portugal 3 Innaxis Foundation & Research Institute, Madrid, Spain 4 Department of Physics and Fundamental Mechanics Applied to Agroforestry Engineering, Universidad Politécnica de Madrid, Madrid, Spain 5 Modeling and Simulation Laboratory, Business Faculty, Universidad del Rosario de Colombia, Bogotá, Colombia 6 Complex Systems Group & GISC, Universidad Rey Juan Carlos, Móstoles, Spain Neuroscientists often assume that the brain is organized as a small-world network, a structure where few connecting links drastically shorten the distance between closely knit groups of nodes. However, the experimental quantification of the small-world structure and its interpretation in terms of information processing are so fraught with technical, to provide a conclusive answer to this question? In a typical experimental setting, neuroscientists record brain images, define nodes and links, construct a network, extract its topological properties, to finally assess their statistical significance and their possible functional meaning. Behind each of these stages, particularly when studying functional Manuscript Click here to download Manuscript: SW 17 06 2015 def.docx 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 rstb.royalsocietypublishing.org Introduction Cite this article: Papo D, Buldu´ JM, Boccaletti S, Bullmore ET. 2014 Complex network theory and the brain. Phil. Trans. R. Soc. B 369: 20130520. http://dx.doi.org/10.1098/rstb.2013.0520 One contribution of 12 to a Theme Issue ‘Complex network theory and the brain’. Subject Areas: cognition, neuroscience Keywords: topology, graph, connectome, neuroimaging, hubs, community structure Author for correspondence: David Papo Complex network theory and the brain David Papo1, Javier M. Buldu´1,2, Stefano Boccaletti3 and Edward T. Bullmore4,5 1 Center for Biomedical Technology, Universidad Polite´cnica de Madrid, Madrid, Spain 2 Complex Systems Group, Universidad Rey Juan Carlos, Mo´stoles, Spain 3 CNR, Istituto dei Sistemi Complessi, Florence, Italy 4 Department of Psychiatry, Behavioural and Clinical Neurosciences Institute, University of Cambridge, Cambridge, UK 5 GlaxoSmithKline, Alternative Discovery and Development, Addenbrooke’s Centre for Clinical Investigations, Cambridge, UK 1. Brain networks: from anatomy to topology The first clear, recognizably scientific representations of the human brain were the drawings and engravings of the Renaissance anatomists. These prototype anatom- ical maps of brain organization demonstrated a physical structure somewhat walnut-like in appearance: an approximately symmetrical pair of deeply wrinkled lobes connected to each other by a central bridge of tissue. More extensive and detailed dissection of the human brain revealed that its convoluted surface is thinly covered (less than 3 mm) by a layer of so-called grey matter— the cortex; and that anatomically separated regions of cortical grey matter are extensively interconnected to each other (and to subcortical grey matter nuclei) by axonal projections that are bundled together to form macroscopically visi- ble white matter tracts, including the major white matter tract linking the two cerebral hemispheres. Even these few fundamental observations on the anatomical organization of the brain indicate that it must be considered as a large-scale (more than 1 mm) net- work of grey matter regions connected by white matter tracts. It has also been increasingly well understood, since the first microscopic neuro-anatomists of the nineteenth century, that there is an intricate pattern of synaptic connections between locally neighbouring neurons in the same cortical column or area. So there has long been strong evidence that the brain has a qualitatively complex network organization at micro (less than 1 mm) as well as macro scales. At a microscopic scale, we know that drawing a complete network diagram of the human brain would be a task of currently unmanageable scale and technical difficulty. The brain comprises an estimated 1011 neurons (105 mm–3 ) and axonal on September 1, 2014rstb.royalsocietypublishing.orgDownloaded from rstb.royalsocietypublishing.org Opinion piece Cite this article: Papo D, Zanin M, Pineda-Pardo JA, Boccaletti S, Buldu´ JM. 2014 Functional brain networks: great expectations, hard times and the big leap forward. Phil. Trans. R. Soc. B 369: 20130525. http://dx.doi.org/10.1098/rstb.2013.0525 One contribution of 12 to a Theme Issue ‘Complex network theory and the brain’. Subject Areas: neuroscience, cognition Keywords: complex networks theory, functional neuroimaging, small-world, robustness, efficiency, synchronizability Author for correspondence: David Papo e-mail: papodav@gmail.com Functional brain networks: great expectations, hard times and the big leap forward David Papo1, Massimiliano Zanin2,3, Jose´ Angel Pineda-Pardo1, Stefano Boccaletti4 and Javier M. Buldu´1,5 1 Center for Biomedical Technology, Universidad Polite´cnica de Madrid, Madrid, Spain 2 Faculdade de Cıˆencias e Tecnologia, Departamento de Engenharia, Electrote´cnica, Universidade Nova de Lisboa, Lisboa, Portugal 3 Innaxis Foundation and Research Institute, Madrid, Spain 4 Istituto dei Sistemi Complessi, CNR, Florence, Italy 5 Complex Systems Group, Universidad Rey Juan Carlos, Mo´stoles, Spain Many physical and biological systems can be studied using complex network theory, a new statistical physics understanding of graph theory. The recent application of complex network theory to the study of functional brain networks has generated great enthusiasm as it allows addressing hitherto non-standard issues in the field, such as efficiency of brain functioning or vulnerability to damage. However, in spite of its high degree of generality, the theory was originally designed to describe systems profoundly different from the brain. We discuss some important caveats in the wholesale application of existing tools and concepts to a field they were not originally designed to describe. At the same time, we argue that complex network theory has not yet been taken full advantage of, as many of its important aspects are yet to make their appearance in the neuroscience literature. Finally, we propose that, rather than simply borrowing from an existing theory, functional neural networks can inspire a fundamental reformulation of complex network theory, to account for its exquisitely complex functioning mode. 1. Introduction Characterizing how the brain organizes its activity to carry out complex tasks is highly non-trivial. While early neuroimaging and electrophysiological studies typically aimed at identifying patches of task-specific activation or local time- varying patterns of activity, there has now been consensus that task-related brain activity has a temporally multiscale, spatially extended character, as net- works of coordinated brain areas are continuously formed and destroyed [1,2]. Up until recently, though, the emphasis of functional brain activity studies has been on the identity of the particular nodes forming these networks, and on the characterization of connectivity metrics between them [3], the underlying covert hypothesis being that each node, constituting a coarse-grained represen- tation of a given brain region, provides a unique contribution to the whole. Thus, functional neuroimaging initially integrated the two basic ingredients of early neuropsychology: localization of cognitive function into specialized brain modules and the role of connection fibres in the integration of various modules. Lately, brain structure and function have started being investigated using complex network theory, a statistical mechanics understanding of an old branch of pure mathematics: graph theory [4]. Graph theory allows endowing networks with a great number of quantitative properties [5,6], thus vastly enriching the set of objective descriptors of brain structure and function at neuroscientists’ disposal. However, in spite of a great potential, the results have so far not entirely met the expectations in that complex network theory has not yet given rise to a on September 1, 2014rstb.royalsocietypublishing.orgDownloaded from OPINION ARTICLE published: 27 February 2014 doi: 10.3389/fnhum.2014.00107 Reconstructing functional brain networks: have we got the basics right? David Papo1 *, Massimiliano Zanin2,3 and Javier M. Buldú4,5 1 Computational Systems Biology Group, Center for Biomedical Technology, Universidad Politécnica de Madrid, Madrid, Spain 2 Departamento de Engenharia Electrotecnica, Faculdade de Ciencias e Tecnologia, Universidade Nova de Lisboa, Lisboa, Portugal 3 Innaxis Foundation & Research Institute, Madrid, Spain 4 Laboratory of Biological Networks, Center for Biomedical Technology, Universidad Politécnica de Madrid, Madrid, Spain 5 Departamento de Tecnología Electrónica, Universidad Rey Juan Carlos, Móstoles, Spain *Correspondence: papodav@gmail.com Edited by: Daniel S. Margulies, Max Planck Institute for Human Cognitive and Brain Sciences, Germany Keywords: complex networks theory, functional brain networks, correlations, synchronization, data mining Both at rest and during the executions of cognitive tasks, the brain continuously cre- ates and reshapes complex patterns of cor- general are defined in system-level studies using noninvasive techniques, which may be critical when interpreting the results of in spatial correlations in the topology of reconstructed networks. Even more importantly, sub-sampling HUMAN NEUROSCIENCE Reorganization of Functional Networks in Mild Cognitive Impairment Javier M. Buldu´ 1,2 *, Ricardo Bajo3 , Fernando Maestu´ 3 , Nazareth Castellanos3 , Inmaculada Leyva1,2 , Pablo Gil4 , Irene Sendin˜ a-Nadal1,2 , Juan A. Almendral1,2 , Angel Nevado3 , Francisco del-Pozo3 , Stefano Boccaletti5,6 1 Complex Systems Group, Universidad Rey Juan Carlos, Fuenlabrada, Spain, 2 Laboratory of Biological Networks, Centre for Biomedical Technology, Madrid, Spain, 3 Cognitive and Computational Neuroscience Lab, Centre for Biomedical Technology, Polytechnic and Complutense University of Madrid (UPM-UCM), Madrid, Spain, 4 Memory Unit, Hospital Clı´nico San Carlos, Madrid, Spain, 5 Computational Systems Biology Group, Centre for Biomedical Technology, Madrid, Spain, 6 Istituto dei Sistemi Complessi, CNR, Florence, Italy Abstract Whether the balance between integration and segregation of information in the brain is damaged in Mild Cognitive Impairment (MCI) subjects is still a matter of debate. Here we characterize the functional network architecture of MCI subjects by means of complex networks analysis. Magnetoencephalograms (MEG) time series obtained during a memory task were evaluated by synchronization likelihood (SL), to quantify the statistical dependence between MEG signals and to obtain the functional networks. Graphs from MCI subjects show an enhancement of the strength of connections, together with an increase in the outreach parameter, suggesting that memory processing in MCI subjects is associated with higher energy expenditure and a tendency toward random structure, which breaks the balance between integration and segregation. All features are reproduced by an evolutionary network model that simulates the degenerative process of a healthy functional network to that associated with MCI. Due to the high rate of conversion from MCI to Alzheimer Disease (AD), these results show that the analysis of functional networks could be an appropriate tool for the early detection of both MCI and AD. Citation: Buldu´ JM, Bajo R, Maestu´ F, Castellanos N, Leyva I, et al. (2011) Reorganization of Functional Networks in Mild Cognitive Impairment. PLoS ONE 6(5): e19584. doi:10.1371/journal.pone.0019584 Editor: Michal Zochowski, University of Michigan, United States of America Received December 17, 2010; Accepted April 1, 2011; Published May 23, 2011 Copyright: ß 2011 Buldu´ et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Funding: This work was supported by MADRI.B project, Obra Social Caja Madrid, by the Spanish Ministry of S&T [FIS2009-07072, PSI2009-14415-C03-01] and by the Community of Madrid under the R&D Program of activities MODELICO-CM [S2009ESP-1691]. All funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests: The authors have declared that no competing interests exist. * E-mail: javier.buldu@urjc.es Author's personal copy Principles of recovery from traumatic brain injury: Reorganization of functional networks Nazareth P. Castellanos a, ⁎, Inmaculada Leyva b,c, ⁎, Javier M. Buldú b,c , Ricardo Bajo a , Nuria Paúl d , Pablo Cuesta a , Victoria E. Ordóñez a , Cristina L. Pascua e , Stefano Boccaletti f , Fernando Maestú a , Francisco del-Pozo a a Cognitive and Computational Neuroscience Laboratory, Centre for Biomedical Technology (CTB), Technical University of Madrid and Complutense University of Madrid, Spain b Complex Systems Group, Universidad Rey Juan Carlos, Fuenlabrada, Spain c Laboratory of Biological Networks, Centre for Biomedical Technology (CTB), Technical University of Madrid, Spain d Department of Psychiatric and Medical Psychology, Medicine School, Complutense University of Madrid, Spain e Centre of Brain Injury Treatment LESCER, Madrid, Spain f CNR-Institute for Complex Systems, Florence, Italy a b s t r a c ta r t i c l e i n f o Article history: Received 19 July 2010 Revised 1 December 2010 Accepted 16 December 2010 Available online 29 December 2010 Keywords: Magnetoencephalography (MEG) Functional connectivity Graph theory Traumatic brain injury (TBI) Plasticity Recovery after brain injury is an excellent platform to study the mechanism underlying brain plasticity, the reorganization of networks. Do complex network measures capture the physiological and cognitive alterations that occurred after a traumatic brain injury and its recovery? Patients as well as control subjects underwent resting-state MEG recording following injury and after neurorehabilitation. Next, network measures such as network strength, path length, efficiency, clustering and energetic cost were calculated. We show that these parameters restore, in many cases, to control ones after recovery, specifically in delta and alpha bands, and we design a model that gives some hints about how the functional networks modify their weights in the recovery process. Positive correlations between complex network measures and some of the general index of the WAIS-III test were found: changes in delta-based path-length and those in Performance IQ score, and alpha-based normalized global efficiency and Perceptual Organization Index. These results indicate that: 1) the principle of recovery depends on the spectral band, 2) the structure of the functional networks evolves in parallel to brain recovery with correlations with neuropsychological scales, and 3) energetic cost reveals an optimal principle of recovery. © 2010 Elsevier Inc. All rights reserved. Introduction Traditionally, localizationist and holist views of brain function have exclusively emphasized either functional segregation or functional integration among components of the nervous system. While segrega- tion indicates a high functional specialization of each brain region, integration highlights the idea of a global structure and cooperative behaviour. Neither of these views alone adequately accounts for the multiple levels at which interactions occur during brain functioning. Modern views conceive the human brain as having the capacity to conjoin local specialization with global integration (Tononi et al., 1994). Under this framework, the study of brain functioning is based on the idea that the brain is a complex network of complex systems with abundant interactions between local and distant areas (Singer, 1999; Varela et al., 2001; Fries, 2005; 2009; Singer, 2009). An approach to understand the dynamical nature of the links between neural assemblies could be functional connectivity (Friston et al., 1994), which refers to the statistical interdependencies between physiological time series recorded in various brain areas (Aertsen et al., 1989). Functional connectivity is, then, an essential tool for the study of brain functioning and the implications of the deviation from healthy patterns is a much debated question recently (Schnitzler and Gross, 2005; Guggisberg et al., 2008). Functional connectivity patterns have been proved to be altered by brain injury but, could they also reflect the capability of brain to compensate for such injury? One could think that it is possible, since brain plasticity produces changes at multiple levels of neuronal reorganization, from synapses to cortical maps and large-scale neuronal networks (Buonomano and Merzenich, 1998). Studies of the changes which occurred in the functional connectivity patterns after brain tumor rejections (Douw et al., 2008), recovery from capsular stroke (Gerloff et al., 2006) or traumatic brain injury (Castellanos et al., 2010) are some examples of the way the brain reorganizes after lesion. However, little is known about the principles governing the structural reorganization of functional networks after an acquired brain injury and during recovery. NeuroImage 55 (2011) 1189–1199 ⁎ Corresponding authors. N.P. Castellanos is to be contacted at Laboratory of Cognitive and Computational Neuroscience, Centre of Biomedical Technology (CTB), Universidad Politécnica de Madrid, Campus de Montegancedo, 28660 Madrid, Spain. I. Leyva, Complex Systems Group, Universidad Rey Juan Carlos, Camino del Molino s/n, 28943 Fuenlabrada, Madrid, Spain. E-mail address: nazareth@pluri.ucm.es (N.P. Castellanos). 1053-8119/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.neuroimage.2010.12.046 Contents lists available at ScienceDirect NeuroImage journal homepage: www.elsevier.com/locate/ynimg ALGUNAS REFERENCIAS AL RESPECTO…

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