2848:
system has been successfully used in a wide variety of neuroscience research studies. In particular, it predicted the existence of spiral waves, which can occur during seizures; this theoretical prediction was subsequently confirmed experimentally using optical imaging of slices from the rat cortex.
2180:
of these issues, developed in 2008 by
Shusterman and Troy using the Wilson–Cowan model, predicts qualitative and quantitative features of epileptiform activity. In particular, it accurately predicts the propagation speed of epileptic seizures (which is approximately 4–7 times slower than normal brain
52:
The Wilson–Cowan model considers a homogeneous population of interconnected neurons of excitatory and inhibitory subtypes. All cells receive the same number of excitatory and inhibitory afferents, that is, all cells receive the same average excitation, x(t). The target is to analyze the evolution in
39:
and extensions of the model have been widely used in modeling neuronal populations. The model is important historically because it uses phase plane methods and numerical solutions to describe the responses of neuronal populations to stimuli. Because the model neurons are simple, only elementary limit
4123:
To summarize, mathematical modeling and theoretical analysis of large-scale electrophysiological activity provide tools for predicting the spread and migration of hypersynchronous brain activity, which can be useful for diagnostic evaluation and management of patients with epilepsy. It might be also
4016:
is a fraction of the traveling wave speed, which is consistent with experimental and clinical observations regarding the slow spread of epileptic activity. This migration mechanism also provides a plausible explanation for spread and sustenance of epileptiform activity without a driving source that,
2189:
The transition from normal state of brain activity to epileptic seizures was not formulated theoretically until 2008, when a theoretical path from a baseline state to large-scale self-sustained oscillations, which spread out uniformly from the point of stimulus, has been mapped for the first time.
2263:
spatially dependent extension of the classical Wilson–Cowan model can be utilized. Under appropriate initial conditions, the excitatory component, u, dominates over the inhibitory component, I, and the three-variable system reduces to the two-variable Pinto-Ermentrout type model
2212:
This baseline state represents activity of the brain in the state of relaxation, in which neurons receive some level of spontaneous, weak stimulation by small, naturally present concentrations of neurohormonal substances. In waking adults this state is commonly associated with
2434:
633:
373:
2147:
4119:
in formula (1) shows that the leading edge of the region of synchronous seizure activity migrates approximately 4–7 times more slowly than normal brain wave activity, which is in agreement with the experimental data described above.
4025:
The rate of migration of hypersynchronous activity that was experimentally recorded during seizures in a human subject, using chronically implanted subdural electrodes on the surface of the left temporal lobe, has been estimated as
1919:
3155:
proportional to the wave speed. From this initial observation it is natural to expect that the proportionality constant should be the fraction of the time that the solution is concave during one cycle." Therefore, when
3352:
4474:
Milton, J. G. (2010). "Mathematical Review: From baseline to epileptiform activity: A path to synchronized rhythmicity in large-scale neural networks by V. Shusterman and W. C. Troy (Phys. Rev. E 77: 061911)".
2500:
1549:
3681:
3964:
1705:, obtaining in this way the product at right side of the first equation of the model (with the assumption of uncorrelated terms). Same rationale can be done for inhibitory cells, obtaining second equation.
1192:
2664:
3604:
1029:
3538:
1658:
1357:
1285:
4010:
3847:
1067:
If, instead of all cells receiving same excitatory inputs and different threshold, we consider that all cells have same threshold but different number of afferent synapses per cell, being
3772:
2773:
2269:
3804:
3187:
3107:
2717:
4076:
2995:
2528:
3707:
2814:
2589:
3733:
3431:
2905:
2554:
1703:
3463:
3042:
1586:
962:
735:
3149:
704:
671:
3377:
44:, and stimulus-dependent evoked responses are predicted. The key findings include the existence of multiple stable states, and hysteresis, in the population response.
2261:
929:
4117:
3015:
3074:
2969:
2937:
2846:
1219:
845:
818:
791:
764:
1094:
905:
876:
109:
80:
1120:
1062:
1380:
4351:
Wright, J. J. (1 August 1999). "Simulation of EEG: dynamic changes in synaptic efficacy, cerebral rhythms, and dissipative and generative activity in cortex".
3109:
where bulk oscillations occur, "The rate of expansion of the hypersynchronization region is determined by an interplay between two key features: (i) the speed
2857:
The expansion of hypersynchronized regions exhibiting large-amplitude stable bulk oscillations occurs when the oscillations coexist with the stable rest state
964:
is the distribution of thresholds in all cells, then the expected proportion of neurons receiving an excitation at or above threshold level per unit time is:
379:
119:
2159:
The determination of three concepts is fundamental to an understanding of hypersynchronization of neurophysiological activity at the global (system) level:
4685:
Kilpatrick, Z. P.; Bressloff, P. C. (2010). "Effects of synaptic depression and adaptation on spatiotemporal dynamics of an excitatory neuronal network".
3076:
system: a stable, spatially independent, periodic solution (bulk oscillation) and a stable rest state coexist over a continuous range of parameters. When
1925:
1722:
4291:
Robinson, P. A.; Rennie, C. J.; Wright, J. J. (1 July 1997). "Propagation and stability of waves of electrical activity in the cerebral cortex".
4544:
Pinto, D.; Ermentrout, G. B. (2001). "Spatially
Structured Activity in Synaptically Coupled Neuronal Networks: I. Traveling Fronts and Pulses".
4413:
Shusterman, V; Troy, WC (2008). "From baseline to epileptiform activity: a path to synchronized rhythmicity in large-scale neural networks".
4205:
Wilson, H. R.; Cowan, J. D. (1 September 1973). "A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue".
3194:
2163:
The mechanism by which normal (baseline) neurophysiological activity evolves into hypersynchronization of large regions of the brain during
2209:
A time-varying component which may include singlepulse waves, multipulse waves, or periodic waves caused by spontaneous neuronal activity.
3113:
of waves that form and propagate outward from the edge of the region, and (ii) the concave shape of the graph of the activation variable
2441:
4884:
3611:
3855:
1127:
2594:
793:
are respectively the connectivity coefficient giving the average number of excitatory and inhibitory synapses per excitatory cell;
4335:"D.T.J. Liley, P.J. Cadusch and J.J. Wright. A continuum theory of electrocortical activity. Neurocomputing. 26–27:795–800 (1999)"
4494:
Wilson, H. R.; Cowan, J. D. (1973). "A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue".
1387:
4869:
Epilepsy as a
Dynamic Disease, edited by J. Milton and P. Jung, Biological and Medical Physics series Springer, Berlin, 2003.
3545:
3151:
due to the concave component, the stable solitary wave emanating from the region causes the region to expand spatially at a
969:
3470:
2203:
A time-independent component represented by subthreshold excitatory activity E and superthreshold inhibitory activity I.
2719:
The firing rate function, which is generally accepted to have a sharply increasing sigmoidal shape, is approximated by
4124:
useful for predicting migration and spread of electrical activity over large regions of the brain that occur during
3972:
3809:
4904:
2225:) rhythms are usually observed during deeper relaxation and sleep. To describe this general setting, a 3-variable
20:
2429:{\displaystyle {\partial u \over \partial t}=u-v+\int _{R^{2}}\omega (x-x',y-y')f(u-\theta )\,dxdy+\zeta (x,y,t),}
4894:
3738:
4899:
2722:
1591:
1290:
2997:
is held fixed. The linearized system exhibits subthreshold decaying oscillations whose frequency increases as
3777:
1224:
3465:
it is necessary to analyze the underlying bulk oscillation which satisfies the spatially independent system
1714:
3159:
3079:
2669:
4553:
27:
describes the dynamics of interactions between populations of very simple excitatory and inhibitory model
4032:
4825:
4739:
4694:
4641:
4593:
4422:
4300:
4257:
4161:
2974:
2507:
706:
are functions of sigmoid form that depends on the distribution of the trigger thresholds (see below)
4558:
3686:
2778:
932:
41:
36:
4816:
Wilson, H. R.; Blake, R.; Lee, S. H. (2001). "Dynamics of travelling waves in visual perception".
2559:
4849:
4519:
4376:
4316:
4230:
3712:
3402:
2860:
2533:
2181:
wave activity) in a human subject with chronically implanted electroencephalographic electrodes.
2177:
1663:
32:
3437:
3020:
1556:
938:
711:
3128:
676:
643:
4889:
4841:
4798:
4667:
4511:
4438:
4368:
4273:
4222:
4187:
3359:
2907:. To understand the mechanism responsible for the expansion, it is necessary to linearize the
2164:
4765:
Huang, X.; Troy, W. C.; Yang, Q.; Ma, H.; Laing, C. R.; Schiff, S. J.; Wu, J. Y. (Nov 2004).
2228:
914:
4833:
4788:
4778:
4747:
4710:
4702:
4657:
4649:
4609:
4601:
4584:
Folias, S. E.; Bressloff, P. C. (2004). "Breathing pulses in an excitatory neural network".
4563:
4503:
4430:
4360:
4308:
4265:
4214:
4177:
4169:
4085:
3387:
is the speed of waves emanating from the hypersynchronization region. A realistic value of
3000:
1035:
3047:
2942:
2910:
2819:
2193:
A realistic state of baseline physiological activity has been defined, using the following
1204:
823:
796:
769:
742:
4248:
Jirsa, V. K.; Haken, H. (29 July 1996). "Field Theory of
Electromagnetic Brain Activity".
1070:
881:
852:
85:
56:
4632:
Folias, S. E.; Bressloff, P. C. (Nov 2005). "Breathers in two-dimensional neural media".
4082:
which is consistent with the theoretically predicted range given above in (2). The ratio
1099:
1041:
4829:
4743:
4698:
4645:
4597:
4426:
4304:
4261:
4165:
628:{\displaystyle I(t+\tau )=\left\;S_{i}\left(\int _{-\infty }^{t}\alpha (t-t')dt'\right)}
368:{\displaystyle E(t+\tau )=\left\;S_{e}\left(\int _{-\infty }^{t}\alpha (t-t')dt'\right)}
4793:
4766:
4182:
4149:
1365:
4173:
1221:
the refractory period after a trigger, the proportion of cells in refractory period is
4878:
4320:
3121:=0 to the activation threshold. Numerical experiments show that during the rise of
4853:
4783:
4380:
3125:
towards threshold, as the rate of vertical increase slows down, over time interval
4653:
4523:
4234:
4150:"Excitatory and inhibitory interactions in localized populations of model neurons"
2142:{\displaystyle \tau '{\frac {d{\bar {I}}}{dt}}=-{\bar {I}}+(1-r'{\bar {I}})S_{i}}
4706:
4334:
1717:
modeling is assumed the model simplifies, being the new equations of the model:
4434:
4269:
1914:{\displaystyle \tau {\frac {d{\bar {E}}}{dt}}=-{\bar {E}}+(1-r{\bar {E}})S_{e}}
4567:
4129:
4125:
2222:
2218:
2214:
2169:
The key factors that govern the rate of expansion of hypersynchronized regions
4715:
4662:
4614:
4312:
4845:
4802:
4671:
4442:
4372:
4277:
1096:
the distribution of the number of afferent synapses, a variant of function
4515:
4364:
4226:
4191:
3044:
where the oscillation frequency is high enough, bistability occurs in the
114:
The equations that describes this evolution are the Wilson-Cowan model:
4507:
4218:
2151:
where bar terms are the time coarse-grained versions of original ones.
4751:
4730:
Laing, C. R.; Troy, W. C. (2003). "PDE methods for non-local models".
4605:
3347:{\displaystyle Rate=(\Delta t/T)*c~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(1)}
4837:
3608:
This system is derived using standard functions and parameter values
3117:
as it rises, during each bulk oscillation cycle, from the rest state
28:
53:
time of number of excitatory and inhibitory cells firing at time t,
4460:
V. L. Towle, F. Ahmad, M. Kohrman, K. Hecox, and S. Chkhenkeli, in
4017:
despite a number of experimental studies, has never been observed.
2495:{\displaystyle {\partial v \over \partial t}=\epsilon (\beta u-v).}
2172:
The electrophysiological activity pattern dynamics on a large-scale
16:
Mathematical model of the interactions between neuronal populations
4539:
4537:
4535:
4533:
2556:
determine the rate of change of recovery. The connection function
3676:{\displaystyle \omega =2.1e^{-\lambda {\sqrt {-(x^{2}+y^{2})}}}}
907:
are the external input to the excitatory/inhibitory populations.
3959:{\displaystyle 3.046mm/s\leq Rate\leq 5.331mm/s~~~~~~~~~~~~(2)}
3806:, Shusterman and Troy analyzed the bulk oscillations and found
1187:{\displaystyle S(x)=\int _{\frac {\theta }{x}}^{\infty }C(w)dw}
2659:{\displaystyle \omega =Ae^{-\lambda {\sqrt {-(x^{2}+y^{2})}}}}
2591:
is positive, continuous, symmetric, and has the typical form
1362:
The average excitation level of an excitatory cell at time
1287:
and the proportion of sensitive (able to trigger) cells is
1709:
Simplification of the model assuming time coarse graining
1544:{\displaystyle x(t)=\int _{-\infty }^{t}\alpha (t-t')dt'}
4132:), cognitive activity and in other functional settings.
3599:{\displaystyle {{dv} \over {dt}}=\epsilon (\beta u-v).}
4021:
Comparing theoretical and experimental migration rates
3189:, the rate of expansion of the region is estimated by
4627:
4625:
4579:
4577:
4088:
4035:
3975:
3858:
3812:
3780:
3741:
3715:
3689:
3614:
3548:
3473:
3440:
3405:
3362:
3197:
3162:
3131:
3082:
3050:
3023:
3003:
2977:
2945:
2913:
2863:
2822:
2781:
2725:
2672:
2597:
2562:
2536:
2510:
2504:
The variable v governs the recovery of excitation u;
2444:
2272:
2231:
1928:
1725:
1666:
1594:
1559:
1553:
Thus, the number of cells that triggers at some time
1390:
1368:
1293:
1227:
1207:
1130:
1102:
1073:
1044:
972:
941:
917:
884:
855:
826:
799:
772:
745:
714:
679:
646:
382:
122:
88:
59:
1024:{\displaystyle S(x)=\int _{0}^{x}D(\theta )d\theta }
4456:
4454:
4452:
3533:{\displaystyle {{du} \over {dt}}=u-v+H(u-\theta ),}
1588:is the number of cells not in refractory interval,
4767:"Spiral waves in disinhibited mammalian neocortex"
4111:
4070:
4004:
3958:
3841:
3798:
3766:
3727:
3701:
3675:
3598:
3532:
3457:
3425:
3371:
3346:
3181:
3143:
3101:
3068:
3036:
3009:
2989:
2963:
2931:
2899:
2840:
2808:
2767:
2711:
2658:
2583:
2548:
2522:
2494:
2428:
2255:
2141:
1913:
1697:
1652:
1580:
1543:
1374:
1351:
1279:
1213:
1186:
1114:
1088:
1056:
1023:
956:
923:
899:
870:
839:
812:
785:
758:
729:
698:
665:
627:
367:
103:
74:
4408:
4406:
4404:
4402:
4400:
4398:
4396:
4394:
4392:
4390:
4865:
4863:
4005:{\displaystyle 0.136\leq \Delta t/T\leq 0.238}
3842:{\displaystyle 0.136\leq \Delta t/T\leq 0.238}
3379:is the length of subthreshold time interval,
8:
1660:AND that have reached the excitatory level,
693:
690:
660:
657:
3767:{\displaystyle \beta \geq \beta ^{*}=12.61}
2775:, where H denotes the Heaviside function;
469:
209:
4792:
4782:
4714:
4661:
4613:
4557:
4181:
4101:
4087:
4060:
4034:
3988:
3974:
3903:
3868:
3857:
3825:
3811:
3779:
3752:
3740:
3714:
3688:
3660:
3647:
3635:
3628:
3613:
3559:
3551:
3549:
3547:
3484:
3476:
3474:
3472:
3447:
3439:
3412:
3404:
3361:
3222:
3196:
3173:
3161:
3130:
3093:
3081:
3049:
3028:
3022:
3002:
2976:
2944:
2912:
2862:
2821:
2780:
2768:{\displaystyle f(u-\theta )=H(u-\theta )}
2724:
2671:
2643:
2630:
2618:
2611:
2596:
2561:
2535:
2509:
2445:
2443:
2383:
2317:
2312:
2273:
2271:
2230:
2093:
2092:
2086:
2051:
2050:
2044:
2023:
2005:
2004:
1973:
1972:
1944:
1943:
1937:
1927:
1870:
1869:
1863:
1833:
1832:
1826:
1810:
1792:
1791:
1765:
1764:
1736:
1735:
1729:
1724:
1671:
1665:
1653:{\displaystyle 1-\int _{t-r}^{t}E(t')dt'}
1616:
1605:
1593:
1558:
1484:
1454:
1418:
1410:
1389:
1367:
1352:{\displaystyle 1-\int _{t-r}^{t}E(t')dt'}
1315:
1304:
1292:
1243:
1232:
1226:
1206:
1160:
1150:
1129:
1101:
1072:
1043:
997:
992:
971:
940:
916:
883:
854:
831:
825:
804:
798:
777:
771:
750:
744:
713:
684:
678:
651:
645:
563:
533:
497:
489:
474:
430:
419:
381:
303:
273:
237:
229:
214:
170:
159:
121:
87:
58:
4140:
3799:{\displaystyle 12.61\leq \beta \leq 17}
1280:{\displaystyle \int _{t-r}^{t}E(t')dt'}
3383:is period of the periodic solution;
3182:{\displaystyle \beta \geq \beta ^{*}}
3102:{\displaystyle \beta \geq \beta ^{*}}
2712:{\displaystyle (A,\lambda )=(2.1,1).}
847:its counterparts for inhibitory cells
7:
2185:Transition into hypersynchronization
4012:, Eq. (1) shows that the migration
4148:Wilson, H.R.; Cowan, J.D. (1972).
3982:
3819:
3441:
3406:
3363:
3216:
3132:
2456:
2448:
2284:
2276:
1414:
1161:
493:
233:
14:
4071:{\displaystyle Rate\approx 4mm/s}
3017:increases. At a critical value
3391:, derived by Wilson et al., is
2816:is a short-time stimulus. This
4784:10.1523/jneurosci.2705-04.2004
3953:
3947:
3666:
3640:
3590:
3575:
3524:
3512:
3341:
3335:
3230:
3213:
3063:
3051:
2990:{\displaystyle \epsilon >0}
2958:
2946:
2926:
2914:
2894:
2882:
2876:
2864:
2835:
2823:
2803:
2785:
2762:
2750:
2741:
2729:
2703:
2691:
2685:
2673:
2649:
2623:
2578:
2566:
2523:{\displaystyle \epsilon >0}
2486:
2471:
2420:
2402:
2380:
2368:
2362:
2328:
2250:
2232:
2136:
2133:
2127:
2110:
2104:
2098:
2068:
2062:
2056:
2029:
2016:
2010:
1987:
1978:
1949:
1908:
1905:
1899:
1887:
1881:
1875:
1850:
1844:
1838:
1816:
1803:
1797:
1779:
1770:
1741:
1692:
1689:
1683:
1677:
1636:
1625:
1575:
1563:
1527:
1524:
1513:
1504:
1493:
1474:
1463:
1447:
1444:
1427:
1400:
1394:
1335:
1324:
1263:
1252:
1175:
1169:
1140:
1134:
1109:
1106:
1083:
1077:
1051:
1048:
1012:
1006:
982:
976:
951:
945:
894:
888:
865:
859:
737:is the stimulus decay function
724:
718:
606:
603:
592:
583:
572:
553:
542:
526:
523:
506:
450:
439:
398:
386:
346:
343:
332:
323:
312:
293:
282:
266:
263:
246:
190:
179:
138:
126:
98:
92:
69:
63:
1:
4654:10.1103/PhysRevLett.95.208107
4462:Epilepsy as a Dynamic Disease
4174:10.1016/s0006-3495(72)86068-5
3735:Bulk oscillations occur when
3702:{\displaystyle \epsilon =0.1}
2809:{\displaystyle \zeta (x,y,t)}
2584:{\displaystyle \omega (x,y)}
4707:10.1016/j.physd.2009.06.003
3728:{\displaystyle \theta =0.1}
3426:{\displaystyle \Delta t/T?}
2900:{\displaystyle (u,v)=(0,0)}
2549:{\displaystyle \beta >0}
1698:{\displaystyle S_{e}(x(t))}
4921:
4885:Computational neuroscience
4435:10.1103/PhysRevE.77.061911
4270:10.1103/PhysRevLett.77.960
3458:{\displaystyle \Delta t/T}
3399:How to evaluate the ratio
3037:{\displaystyle \beta ^{*}}
1581:{\displaystyle E(t+\tau )}
957:{\displaystyle D(\theta )}
730:{\displaystyle \alpha (t)}
21:computational neuroscience
4568:10.1137/s0036139900346453
3144:{\displaystyle \Delta t,}
699:{\displaystyle S_{i}\{\}}
666:{\displaystyle S_{e}\{\}}
3434:To determine values for
3372:{\displaystyle \Delta t}
2195:two-component definition
48:Mathematical description
4732:SIAM J. Appl. Dyn. Syst
4586:SIAM J. Appl. Dyn. Syst
4313:10.1103/PhysRevE.56.826
4250:Physical Review Letters
3849:. This gives the range
2256:{\displaystyle (u,I,v)}
2155:Application to epilepsy
1197:Derivation of the model
924:{\displaystyle \theta }
4353:Biological Cybernetics
4113:
4112:{\displaystyle Rate/c}
4072:
4006:
3960:
3843:
3800:
3768:
3729:
3703:
3677:
3600:
3534:
3459:
3427:
3373:
3348:
3183:
3145:
3103:
3070:
3038:
3011:
3010:{\displaystyle \beta }
2991:
2965:
2933:
2901:
2842:
2810:
2769:
2713:
2660:
2585:
2550:
2524:
2496:
2430:
2257:
2143:
1915:
1699:
1654:
1582:
1545:
1376:
1353:
1281:
1215:
1188:
1116:
1090:
1058:
1034:that is a function of
1025:
958:
925:
901:
872:
841:
814:
787:
760:
731:
700:
667:
629:
369:
105:
76:
31:. It was developed by
4365:10.1007/s004220050550
4114:
4073:
4007:
3961:
3844:
3801:
3769:
3730:
3704:
3678:
3601:
3535:
3460:
3428:
3374:
3349:
3184:
3146:
3104:
3071:
3069:{\displaystyle (u,v)}
3039:
3012:
2992:
2966:
2964:{\displaystyle (0,0)}
2934:
2932:{\displaystyle (u,v)}
2902:
2843:
2841:{\displaystyle (u,v)}
2811:
2770:
2714:
2661:
2586:
2551:
2525:
2497:
2431:
2258:
2144:
1916:
1700:
1655:
1583:
1546:
1377:
1354:
1282:
1216:
1214:{\displaystyle \tau }
1189:
1117:
1091:
1059:
1026:
959:
926:
902:
873:
842:
840:{\displaystyle c_{4}}
815:
813:{\displaystyle c_{3}}
788:
786:{\displaystyle c_{2}}
761:
759:{\displaystyle c_{1}}
732:
701:
668:
630:
370:
106:
77:
40:cycle behavior, i.e.
4086:
4033:
3973:
3856:
3810:
3778:
3739:
3713:
3687:
3612:
3546:
3471:
3438:
3403:
3360:
3195:
3160:
3129:
3080:
3048:
3021:
3001:
2975:
2943:
2911:
2861:
2820:
2779:
2723:
2670:
2595:
2560:
2534:
2508:
2442:
2270:
2229:
1926:
1723:
1664:
1592:
1557:
1388:
1366:
1291:
1225:
1205:
1128:
1100:
1089:{\displaystyle C(w)}
1071:
1042:
970:
939:
915:
900:{\displaystyle Q(t)}
882:
871:{\displaystyle P(t)}
853:
824:
797:
770:
743:
712:
677:
644:
380:
120:
104:{\displaystyle I(t)}
86:
75:{\displaystyle E(t)}
57:
4830:2001Natur.412..907W
4744:2003SJADS...2..487L
4699:2010PhyD..239..547K
4646:2005PhRvL..95t8107F
4598:2004SJADS...3..378F
4427:2008PhRvE..77f1911S
4305:1997PhRvE..56..826R
4262:1996PhRvL..77..960J
4166:1972BpJ....12....1W
1621:
1423:
1320:
1248:
1165:
1115:{\displaystyle S()}
1057:{\displaystyle D()}
1002:
933:threshold potential
502:
435:
242:
175:
42:neural oscillations
4546:SIAM J. Appl. Math
4508:10.1007/BF00288786
4219:10.1007/BF00288786
4109:
4068:
4002:
3956:
3839:
3796:
3764:
3725:
3699:
3673:
3596:
3530:
3455:
3423:
3369:
3344:
3179:
3141:
3099:
3066:
3034:
3007:
2987:
2961:
2929:
2897:
2838:
2806:
2765:
2709:
2656:
2581:
2546:
2520:
2492:
2426:
2253:
2217:, whereas slower (
2178:canonical analysis
2165:epileptic seizures
2139:
1911:
1695:
1650:
1601:
1578:
1541:
1406:
1372:
1349:
1300:
1277:
1228:
1211:
1184:
1146:
1112:
1086:
1054:
1021:
988:
954:
921:
897:
868:
837:
810:
783:
756:
727:
696:
663:
625:
485:
415:
365:
225:
155:
101:
72:
25:Wilson–Cowan model
4905:Electrophysiology
4824:(6850): 907–910.
4752:10.1137/030600040
4606:10.1137/030602629
4415:Physical Review E
4293:Physical Review E
3946:
3943:
3940:
3937:
3934:
3931:
3928:
3925:
3922:
3919:
3916:
3913:
3669:
3567:
3492:
3395:=22.4 mm/s.
3334:
3331:
3328:
3325:
3322:
3319:
3316:
3313:
3310:
3307:
3304:
3301:
3298:
3295:
3292:
3289:
3286:
3283:
3280:
3277:
3274:
3271:
3268:
3265:
3262:
3259:
3256:
3253:
3250:
3247:
3244:
3241:
2853:Rate of expansion
2652:
2463:
2291:
2101:
2059:
2013:
1981:
1964:
1952:
1878:
1841:
1800:
1773:
1756:
1744:
1375:{\displaystyle t}
1158:
931:denotes a cell's
4912:
4895:Neural circuitry
4870:
4867:
4858:
4857:
4838:10.1038/35091066
4813:
4807:
4806:
4796:
4786:
4777:(44): 9897–902.
4762:
4756:
4755:
4727:
4721:
4720:
4718:
4682:
4676:
4675:
4665:
4629:
4620:
4619:
4617:
4581:
4572:
4571:
4561:
4541:
4528:
4527:
4491:
4485:
4484:
4471:
4465:
4458:
4447:
4446:
4410:
4385:
4384:
4348:
4342:
4341:
4339:
4331:
4325:
4324:
4288:
4282:
4281:
4245:
4239:
4238:
4202:
4196:
4195:
4185:
4145:
4118:
4116:
4115:
4110:
4105:
4077:
4075:
4074:
4069:
4064:
4011:
4009:
4008:
4003:
3992:
3965:
3963:
3962:
3957:
3944:
3941:
3938:
3935:
3932:
3929:
3926:
3923:
3920:
3917:
3914:
3911:
3907:
3872:
3848:
3846:
3845:
3840:
3829:
3805:
3803:
3802:
3797:
3773:
3771:
3770:
3765:
3757:
3756:
3734:
3732:
3731:
3726:
3708:
3706:
3705:
3700:
3682:
3680:
3679:
3674:
3672:
3671:
3670:
3665:
3664:
3652:
3651:
3636:
3605:
3603:
3602:
3597:
3568:
3566:
3558:
3550:
3539:
3537:
3536:
3531:
3493:
3491:
3483:
3475:
3464:
3462:
3461:
3456:
3451:
3432:
3430:
3429:
3424:
3416:
3378:
3376:
3375:
3370:
3353:
3351:
3350:
3345:
3332:
3329:
3326:
3323:
3320:
3317:
3314:
3311:
3308:
3305:
3302:
3299:
3296:
3293:
3290:
3287:
3284:
3281:
3278:
3275:
3272:
3269:
3266:
3263:
3260:
3257:
3254:
3251:
3248:
3245:
3242:
3239:
3226:
3188:
3186:
3185:
3180:
3178:
3177:
3150:
3148:
3147:
3142:
3108:
3106:
3105:
3100:
3098:
3097:
3075:
3073:
3072:
3067:
3043:
3041:
3040:
3035:
3033:
3032:
3016:
3014:
3013:
3008:
2996:
2994:
2993:
2988:
2970:
2968:
2967:
2962:
2938:
2936:
2935:
2930:
2906:
2904:
2903:
2898:
2847:
2845:
2844:
2839:
2815:
2813:
2812:
2807:
2774:
2772:
2771:
2766:
2718:
2716:
2715:
2710:
2665:
2663:
2662:
2657:
2655:
2654:
2653:
2648:
2647:
2635:
2634:
2619:
2590:
2588:
2587:
2582:
2555:
2553:
2552:
2547:
2529:
2527:
2526:
2521:
2501:
2499:
2498:
2493:
2464:
2462:
2454:
2446:
2435:
2433:
2432:
2427:
2361:
2344:
2324:
2323:
2322:
2321:
2292:
2290:
2282:
2274:
2262:
2260:
2259:
2254:
2148:
2146:
2145:
2140:
2123:
2103:
2102:
2094:
2091:
2090:
2081:
2061:
2060:
2052:
2049:
2048:
2039:
2028:
2027:
2015:
2014:
2006:
2003:
1983:
1982:
1974:
1965:
1963:
1955:
1954:
1953:
1945:
1938:
1936:
1920:
1918:
1917:
1912:
1880:
1879:
1871:
1868:
1867:
1843:
1842:
1834:
1831:
1830:
1815:
1814:
1802:
1801:
1793:
1775:
1774:
1766:
1757:
1755:
1747:
1746:
1745:
1737:
1730:
1704:
1702:
1701:
1696:
1676:
1675:
1659:
1657:
1656:
1651:
1649:
1635:
1620:
1615:
1587:
1585:
1584:
1579:
1550:
1548:
1547:
1542:
1540:
1523:
1503:
1489:
1488:
1473:
1459:
1458:
1443:
1422:
1417:
1381:
1379:
1378:
1373:
1358:
1356:
1355:
1350:
1348:
1334:
1319:
1314:
1286:
1284:
1283:
1278:
1276:
1262:
1247:
1242:
1220:
1218:
1217:
1212:
1201:If we denote by
1193:
1191:
1190:
1185:
1164:
1159:
1151:
1121:
1119:
1118:
1113:
1095:
1093:
1092:
1087:
1063:
1061:
1060:
1055:
1030:
1028:
1027:
1022:
1001:
996:
963:
961:
960:
955:
930:
928:
927:
922:
906:
904:
903:
898:
877:
875:
874:
869:
846:
844:
843:
838:
836:
835:
819:
817:
816:
811:
809:
808:
792:
790:
789:
784:
782:
781:
765:
763:
762:
757:
755:
754:
736:
734:
733:
728:
705:
703:
702:
697:
689:
688:
672:
670:
669:
664:
656:
655:
634:
632:
631:
626:
624:
620:
619:
602:
582:
568:
567:
552:
538:
537:
522:
501:
496:
479:
478:
468:
464:
463:
449:
434:
429:
374:
372:
371:
366:
364:
360:
359:
342:
322:
308:
307:
292:
278:
277:
262:
241:
236:
219:
218:
208:
204:
203:
189:
174:
169:
110:
108:
107:
102:
81:
79:
78:
73:
4920:
4919:
4915:
4914:
4913:
4911:
4910:
4909:
4900:Neurophysiology
4875:
4874:
4873:
4868:
4861:
4815:
4814:
4810:
4764:
4763:
4759:
4729:
4728:
4724:
4684:
4683:
4679:
4634:Phys. Rev. Lett
4631:
4630:
4623:
4583:
4582:
4575:
4543:
4542:
4531:
4493:
4492:
4488:
4473:
4472:
4468:
4459:
4450:
4412:
4411:
4388:
4350:
4349:
4345:
4337:
4333:
4332:
4328:
4290:
4289:
4285:
4247:
4246:
4242:
4204:
4203:
4199:
4147:
4146:
4142:
4138:
4084:
4083:
4081:
4031:
4030:
4029:
4023:
3971:
3970:
3968:
3854:
3853:
3852:
3808:
3807:
3776:
3775:
3748:
3737:
3736:
3711:
3710:
3685:
3684:
3656:
3643:
3624:
3610:
3609:
3544:
3543:
3542:
3469:
3468:
3436:
3435:
3401:
3400:
3358:
3357:
3193:
3192:
3169:
3158:
3157:
3127:
3126:
3089:
3078:
3077:
3046:
3045:
3024:
3019:
3018:
2999:
2998:
2973:
2972:
2941:
2940:
2909:
2908:
2859:
2858:
2855:
2818:
2817:
2777:
2776:
2721:
2720:
2668:
2667:
2639:
2626:
2607:
2593:
2592:
2558:
2557:
2532:
2531:
2506:
2505:
2455:
2447:
2440:
2439:
2438:
2354:
2337:
2313:
2308:
2283:
2275:
2268:
2267:
2227:
2226:
2187:
2157:
2116:
2082:
2074:
2040:
2032:
2019:
1996:
1956:
1939:
1929:
1924:
1923:
1859:
1822:
1806:
1748:
1731:
1721:
1720:
1711:
1667:
1662:
1661:
1642:
1628:
1590:
1589:
1555:
1554:
1533:
1516:
1496:
1480:
1466:
1450:
1436:
1386:
1385:
1364:
1363:
1341:
1327:
1289:
1288:
1269:
1255:
1223:
1222:
1203:
1202:
1199:
1126:
1125:
1098:
1097:
1069:
1068:
1040:
1039:
968:
967:
937:
936:
913:
912:
880:
879:
851:
850:
827:
822:
821:
800:
795:
794:
773:
768:
767:
746:
741:
740:
710:
709:
680:
675:
674:
647:
642:
641:
612:
595:
575:
559:
545:
529:
515:
484:
480:
470:
456:
442:
408:
404:
378:
377:
352:
335:
315:
299:
285:
269:
255:
224:
220:
210:
196:
182:
148:
144:
118:
117:
84:
83:
55:
54:
50:
17:
12:
11:
5:
4918:
4916:
4908:
4907:
4902:
4897:
4892:
4887:
4877:
4876:
4872:
4871:
4859:
4808:
4757:
4738:(3): 487–516.
4722:
4716:10044/1/107212
4693:(9): 547–560.
4677:
4663:10044/1/107604
4640:(20): 208107.
4621:
4615:10044/1/107913
4592:(3): 378–407.
4573:
4559:10.1.1.16.6344
4529:
4486:
4466:
4448:
4386:
4359:(2): 131–147.
4343:
4326:
4299:(1): 826–840.
4283:
4256:(5): 960–963.
4240:
4197:
4139:
4137:
4134:
4108:
4104:
4100:
4097:
4094:
4091:
4067:
4063:
4059:
4056:
4053:
4050:
4047:
4044:
4041:
4038:
4022:
4019:
4001:
3998:
3995:
3991:
3987:
3984:
3981:
3978:
3955:
3952:
3949:
3910:
3906:
3902:
3899:
3896:
3893:
3890:
3887:
3884:
3881:
3878:
3875:
3871:
3867:
3864:
3861:
3838:
3835:
3832:
3828:
3824:
3821:
3818:
3815:
3795:
3792:
3789:
3786:
3783:
3763:
3760:
3755:
3751:
3747:
3744:
3724:
3721:
3718:
3698:
3695:
3692:
3668:
3663:
3659:
3655:
3650:
3646:
3642:
3639:
3634:
3631:
3627:
3623:
3620:
3617:
3595:
3592:
3589:
3586:
3583:
3580:
3577:
3574:
3571:
3565:
3562:
3557:
3554:
3529:
3526:
3523:
3520:
3517:
3514:
3511:
3508:
3505:
3502:
3499:
3496:
3490:
3487:
3482:
3479:
3454:
3450:
3446:
3443:
3422:
3419:
3415:
3411:
3408:
3368:
3365:
3343:
3340:
3337:
3238:
3235:
3232:
3229:
3225:
3221:
3218:
3215:
3212:
3209:
3206:
3203:
3200:
3176:
3172:
3168:
3165:
3140:
3137:
3134:
3096:
3092:
3088:
3085:
3065:
3062:
3059:
3056:
3053:
3031:
3027:
3006:
2986:
2983:
2980:
2960:
2957:
2954:
2951:
2948:
2939:system around
2928:
2925:
2922:
2919:
2916:
2896:
2893:
2890:
2887:
2884:
2881:
2878:
2875:
2872:
2869:
2866:
2854:
2851:
2837:
2834:
2831:
2828:
2825:
2805:
2802:
2799:
2796:
2793:
2790:
2787:
2784:
2764:
2761:
2758:
2755:
2752:
2749:
2746:
2743:
2740:
2737:
2734:
2731:
2728:
2708:
2705:
2702:
2699:
2696:
2693:
2690:
2687:
2684:
2681:
2678:
2675:
2651:
2646:
2642:
2638:
2633:
2629:
2625:
2622:
2617:
2614:
2610:
2606:
2603:
2600:
2580:
2577:
2574:
2571:
2568:
2565:
2545:
2542:
2539:
2519:
2516:
2513:
2491:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2467:
2461:
2458:
2453:
2450:
2425:
2422:
2419:
2416:
2413:
2410:
2407:
2404:
2401:
2398:
2395:
2392:
2389:
2386:
2382:
2379:
2376:
2373:
2370:
2367:
2364:
2360:
2357:
2353:
2350:
2347:
2343:
2340:
2336:
2333:
2330:
2327:
2320:
2316:
2311:
2307:
2304:
2301:
2298:
2295:
2289:
2286:
2281:
2278:
2252:
2249:
2246:
2243:
2240:
2237:
2234:
2186:
2183:
2174:
2173:
2170:
2167:
2156:
2153:
2138:
2135:
2132:
2129:
2126:
2122:
2119:
2115:
2112:
2109:
2106:
2100:
2097:
2089:
2085:
2080:
2077:
2073:
2070:
2067:
2064:
2058:
2055:
2047:
2043:
2038:
2035:
2031:
2026:
2022:
2018:
2012:
2009:
2002:
1999:
1995:
1992:
1989:
1986:
1980:
1977:
1971:
1968:
1962:
1959:
1951:
1948:
1942:
1935:
1932:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1886:
1883:
1877:
1874:
1866:
1862:
1858:
1855:
1852:
1849:
1846:
1840:
1837:
1829:
1825:
1821:
1818:
1813:
1809:
1805:
1799:
1796:
1790:
1787:
1784:
1781:
1778:
1772:
1769:
1763:
1760:
1754:
1751:
1743:
1740:
1734:
1728:
1715:coarse-grained
1710:
1707:
1694:
1691:
1688:
1685:
1682:
1679:
1674:
1670:
1648:
1645:
1641:
1638:
1634:
1631:
1627:
1624:
1619:
1614:
1611:
1608:
1604:
1600:
1597:
1577:
1574:
1571:
1568:
1565:
1562:
1539:
1536:
1532:
1529:
1526:
1522:
1519:
1515:
1512:
1509:
1506:
1502:
1499:
1495:
1492:
1487:
1483:
1479:
1476:
1472:
1469:
1465:
1462:
1457:
1453:
1449:
1446:
1442:
1439:
1435:
1432:
1429:
1426:
1421:
1416:
1413:
1409:
1405:
1402:
1399:
1396:
1393:
1371:
1347:
1344:
1340:
1337:
1333:
1330:
1326:
1323:
1318:
1313:
1310:
1307:
1303:
1299:
1296:
1275:
1272:
1268:
1265:
1261:
1258:
1254:
1251:
1246:
1241:
1238:
1235:
1231:
1210:
1198:
1195:
1183:
1180:
1177:
1174:
1171:
1168:
1163:
1157:
1154:
1149:
1145:
1142:
1139:
1136:
1133:
1122:must be used:
1111:
1108:
1105:
1085:
1082:
1079:
1076:
1053:
1050:
1047:
1020:
1017:
1014:
1011:
1008:
1005:
1000:
995:
991:
987:
984:
981:
978:
975:
953:
950:
947:
944:
920:
909:
908:
896:
893:
890:
887:
867:
864:
861:
858:
848:
834:
830:
807:
803:
780:
776:
753:
749:
738:
726:
723:
720:
717:
707:
695:
692:
687:
683:
662:
659:
654:
650:
623:
618:
615:
611:
608:
605:
601:
598:
594:
591:
588:
585:
581:
578:
574:
571:
566:
562:
558:
555:
551:
548:
544:
541:
536:
532:
528:
525:
521:
518:
514:
511:
508:
505:
500:
495:
492:
488:
483:
477:
473:
467:
462:
459:
455:
452:
448:
445:
441:
438:
433:
428:
425:
422:
418:
414:
411:
407:
403:
400:
397:
394:
391:
388:
385:
363:
358:
355:
351:
348:
345:
341:
338:
334:
331:
328:
325:
321:
318:
314:
311:
306:
302:
298:
295:
291:
288:
284:
281:
276:
272:
268:
265:
261:
258:
254:
251:
248:
245:
240:
235:
232:
228:
223:
217:
213:
207:
202:
199:
195:
192:
188:
185:
181:
178:
173:
168:
165:
162:
158:
154:
151:
147:
143:
140:
137:
134:
131:
128:
125:
111:respectively.
100:
97:
94:
91:
71:
68:
65:
62:
49:
46:
33:Hugh R. Wilson
15:
13:
10:
9:
6:
4:
3:
2:
4917:
4906:
4903:
4901:
4898:
4896:
4893:
4891:
4888:
4886:
4883:
4882:
4880:
4866:
4864:
4860:
4855:
4851:
4847:
4843:
4839:
4835:
4831:
4827:
4823:
4819:
4812:
4809:
4804:
4800:
4795:
4790:
4785:
4780:
4776:
4772:
4768:
4761:
4758:
4753:
4749:
4745:
4741:
4737:
4733:
4726:
4723:
4717:
4712:
4708:
4704:
4700:
4696:
4692:
4688:
4681:
4678:
4673:
4669:
4664:
4659:
4655:
4651:
4647:
4643:
4639:
4635:
4628:
4626:
4622:
4616:
4611:
4607:
4603:
4599:
4595:
4591:
4587:
4580:
4578:
4574:
4569:
4565:
4560:
4555:
4551:
4547:
4540:
4538:
4536:
4534:
4530:
4525:
4521:
4517:
4513:
4509:
4505:
4501:
4497:
4490:
4487:
4482:
4478:
4470:
4467:
4463:
4457:
4455:
4453:
4449:
4444:
4440:
4436:
4432:
4428:
4424:
4421:(6): 061911.
4420:
4416:
4409:
4407:
4405:
4403:
4401:
4399:
4397:
4395:
4393:
4391:
4387:
4382:
4378:
4374:
4370:
4366:
4362:
4358:
4354:
4347:
4344:
4336:
4330:
4327:
4322:
4318:
4314:
4310:
4306:
4302:
4298:
4294:
4287:
4284:
4279:
4275:
4271:
4267:
4263:
4259:
4255:
4251:
4244:
4241:
4236:
4232:
4228:
4224:
4220:
4216:
4212:
4208:
4201:
4198:
4193:
4189:
4184:
4179:
4175:
4171:
4167:
4163:
4159:
4155:
4151:
4144:
4141:
4135:
4133:
4131:
4127:
4121:
4106:
4102:
4098:
4095:
4092:
4089:
4079:
4065:
4061:
4057:
4054:
4051:
4048:
4045:
4042:
4039:
4036:
4027:
4020:
4018:
4015:
3999:
3996:
3993:
3989:
3985:
3979:
3976:
3966:
3950:
3908:
3904:
3900:
3897:
3894:
3891:
3888:
3885:
3882:
3879:
3876:
3873:
3869:
3865:
3862:
3859:
3850:
3836:
3833:
3830:
3826:
3822:
3816:
3813:
3793:
3790:
3787:
3784:
3781:
3761:
3758:
3753:
3749:
3745:
3742:
3722:
3719:
3716:
3696:
3693:
3690:
3661:
3657:
3653:
3648:
3644:
3637:
3632:
3629:
3625:
3621:
3618:
3615:
3606:
3593:
3587:
3584:
3581:
3578:
3572:
3569:
3563:
3560:
3555:
3552:
3540:
3527:
3521:
3518:
3515:
3509:
3506:
3503:
3500:
3497:
3494:
3488:
3485:
3480:
3477:
3466:
3452:
3448:
3444:
3433:
3420:
3417:
3413:
3409:
3396:
3394:
3390:
3386:
3382:
3366:
3354:
3338:
3236:
3233:
3227:
3223:
3219:
3210:
3207:
3204:
3201:
3198:
3190:
3174:
3170:
3166:
3163:
3154:
3138:
3135:
3124:
3120:
3116:
3112:
3094:
3090:
3086:
3083:
3060:
3057:
3054:
3029:
3025:
3004:
2984:
2981:
2978:
2955:
2952:
2949:
2923:
2920:
2917:
2891:
2888:
2885:
2879:
2873:
2870:
2867:
2852:
2850:
2832:
2829:
2826:
2800:
2797:
2794:
2791:
2788:
2782:
2759:
2756:
2753:
2747:
2744:
2738:
2735:
2732:
2726:
2706:
2700:
2697:
2694:
2688:
2682:
2679:
2676:
2644:
2640:
2636:
2631:
2627:
2620:
2615:
2612:
2608:
2604:
2601:
2598:
2575:
2572:
2569:
2563:
2543:
2540:
2537:
2517:
2514:
2511:
2502:
2489:
2483:
2480:
2477:
2474:
2468:
2465:
2459:
2451:
2436:
2423:
2417:
2414:
2411:
2408:
2405:
2399:
2396:
2393:
2390:
2387:
2384:
2377:
2374:
2371:
2365:
2358:
2355:
2351:
2348:
2345:
2341:
2338:
2334:
2331:
2325:
2318:
2314:
2309:
2305:
2302:
2299:
2296:
2293:
2287:
2279:
2265:
2247:
2244:
2241:
2238:
2235:
2224:
2220:
2216:
2210:
2208:
2204:
2202:
2198:
2196:
2191:
2184:
2182:
2179:
2171:
2168:
2166:
2162:
2161:
2160:
2154:
2152:
2149:
2130:
2124:
2120:
2117:
2113:
2107:
2095:
2087:
2083:
2078:
2075:
2071:
2065:
2053:
2045:
2041:
2036:
2033:
2024:
2020:
2007:
2000:
1997:
1993:
1990:
1984:
1975:
1969:
1966:
1960:
1957:
1946:
1940:
1933:
1930:
1921:
1902:
1896:
1893:
1890:
1884:
1872:
1864:
1860:
1856:
1853:
1847:
1835:
1827:
1823:
1819:
1811:
1807:
1794:
1788:
1785:
1782:
1776:
1767:
1761:
1758:
1752:
1749:
1738:
1732:
1726:
1718:
1716:
1708:
1706:
1686:
1680:
1672:
1668:
1646:
1643:
1639:
1632:
1629:
1622:
1617:
1612:
1609:
1606:
1602:
1598:
1595:
1572:
1569:
1566:
1560:
1551:
1537:
1534:
1530:
1520:
1517:
1510:
1507:
1500:
1497:
1490:
1485:
1481:
1477:
1470:
1467:
1460:
1455:
1451:
1440:
1437:
1433:
1430:
1424:
1419:
1411:
1407:
1403:
1397:
1391:
1383:
1369:
1360:
1345:
1342:
1338:
1331:
1328:
1321:
1316:
1311:
1308:
1305:
1301:
1297:
1294:
1273:
1270:
1266:
1259:
1256:
1249:
1244:
1239:
1236:
1233:
1229:
1208:
1196:
1194:
1181:
1178:
1172:
1166:
1155:
1152:
1147:
1143:
1137:
1131:
1123:
1103:
1080:
1074:
1065:
1064:is unimodal.
1045:
1037:
1032:
1018:
1015:
1009:
1003:
998:
993:
989:
985:
979:
973:
965:
948:
942:
934:
918:
891:
885:
862:
856:
849:
832:
828:
805:
801:
778:
774:
751:
747:
739:
721:
715:
708:
685:
681:
652:
648:
640:
639:
638:
635:
621:
616:
613:
609:
599:
596:
589:
586:
579:
576:
569:
564:
560:
556:
549:
546:
539:
534:
530:
519:
516:
512:
509:
503:
498:
490:
486:
481:
475:
471:
465:
460:
457:
453:
446:
443:
436:
431:
426:
423:
420:
416:
412:
409:
405:
401:
395:
392:
389:
383:
375:
361:
356:
353:
349:
339:
336:
329:
326:
319:
316:
309:
304:
300:
296:
289:
286:
279:
274:
270:
259:
256:
252:
249:
243:
238:
230:
226:
221:
215:
211:
205:
200:
197:
193:
186:
183:
176:
171:
166:
163:
160:
156:
152:
149:
145:
141:
135:
132:
129:
123:
115:
112:
95:
89:
66:
60:
47:
45:
43:
38:
37:Jack D. Cowan
34:
30:
26:
22:
4821:
4817:
4811:
4774:
4770:
4760:
4735:
4731:
4725:
4690:
4686:
4680:
4637:
4633:
4589:
4585:
4549:
4545:
4502:(2): 55–80.
4499:
4495:
4489:
4480:
4476:
4469:
4461:
4418:
4414:
4356:
4352:
4346:
4329:
4296:
4292:
4286:
4253:
4249:
4243:
4213:(2): 55–80.
4210:
4206:
4200:
4157:
4153:
4143:
4122:
4080:
4028:
4024:
4013:
3967:
3851:
3607:
3541:
3467:
3398:
3397:
3392:
3388:
3384:
3380:
3355:
3191:
3152:
3122:
3118:
3114:
3110:
2856:
2503:
2437:
2266:
2215:alpha rhythm
2211:
2206:
2205:
2200:
2199:
2194:
2192:
2188:
2175:
2158:
2150:
1922:
1719:
1712:
1552:
1384:
1361:
1200:
1124:
1066:
1033:
966:
910:
636:
376:
116:
113:
51:
24:
18:
4771:J. Neurosci
4464:, pp. 69–81
4160:(1): 1–24.
4879:Categories
4496:Kybernetik
4207:Kybernetik
4154:Biophys. J
4136:References
4130:Delta wave
4126:deep sleep
2666:. In Ref.
1713:When time
4687:Physica D
4554:CiteSeerX
4477:Math. Rev
4321:121138051
4049:≈
3997:≤
3983:Δ
3980:≤
3892:≤
3877:≤
3834:≤
3820:Δ
3817:≤
3791:≤
3788:β
3785:≤
3754:∗
3750:β
3746:≥
3743:β
3717:θ
3691:ϵ
3638:−
3633:λ
3630:−
3616:ω
3585:−
3579:β
3573:ϵ
3522:θ
3519:−
3501:−
3442:Δ
3407:Δ
3364:Δ
3234:∗
3217:Δ
3175:∗
3171:β
3167:≥
3164:β
3133:Δ
3095:∗
3091:β
3087:≥
3084:β
3030:∗
3026:β
3005:β
2979:ϵ
2783:ζ
2760:θ
2757:−
2739:θ
2736:−
2683:λ
2621:−
2616:λ
2613:−
2599:ω
2564:ω
2538:β
2512:ϵ
2481:−
2475:β
2469:ϵ
2457:∂
2449:∂
2400:ζ
2378:θ
2375:−
2352:−
2335:−
2326:ω
2310:∫
2300:−
2285:∂
2277:∂
2099:¯
2072:−
2057:¯
2011:¯
1994:−
1979:¯
1970:−
1950:¯
1931:τ
1876:¯
1854:−
1839:¯
1798:¯
1786:−
1771:¯
1762:−
1742:¯
1727:τ
1610:−
1603:∫
1599:−
1573:τ
1478:−
1434:−
1425:α
1415:∞
1412:−
1408:∫
1309:−
1302:∫
1298:−
1237:−
1230:∫
1209:τ
1162:∞
1153:θ
1148:∫
1019:θ
1010:θ
990:∫
949:θ
919:θ
716:α
637:where:
557:−
513:−
504:α
494:∞
491:−
487:∫
424:−
417:∫
413:−
396:τ
297:−
253:−
244:α
234:∞
231:−
227:∫
164:−
157:∫
153:−
136:τ
4890:Epilepsy
4846:11528478
4803:15525774
4672:16384107
4483:: 92025.
4443:18643304
4373:10481241
4278:10062950
2359:′
2342:′
2121:′
2079:′
2037:′
2001:′
1934:′
1647:′
1633:′
1538:′
1521:′
1501:′
1471:′
1441:′
1346:′
1332:′
1274:′
1260:′
1038:form if
617:′
600:′
580:′
550:′
520:′
461:′
447:′
357:′
340:′
320:′
290:′
260:′
201:′
187:′
4854:4431136
4826:Bibcode
4794:4413915
4740:Bibcode
4695:Bibcode
4642:Bibcode
4594:Bibcode
4552:: 206.
4516:4767470
4423:Bibcode
4381:6558413
4301:Bibcode
4258:Bibcode
4227:4767470
4192:4332108
4183:1484078
4162:Bibcode
3774:. When
1036:sigmoid
29:neurons
4852:
4844:
4818:Nature
4801:
4791:
4670:
4556:
4524:292546
4522:
4514:
4441:
4379:
4371:
4319:
4276:
4235:292546
4233:
4225:
4190:
4180:
3969:Since
3945:
3942:
3939:
3936:
3933:
3930:
3927:
3924:
3921:
3918:
3915:
3912:
3356:where
3333:
3330:
3327:
3324:
3321:
3318:
3315:
3312:
3309:
3306:
3303:
3300:
3297:
3294:
3291:
3288:
3285:
3282:
3279:
3276:
3273:
3270:
3267:
3264:
3261:
3258:
3255:
3252:
3249:
3246:
3243:
3240:
23:, the
4850:S2CID
4520:S2CID
4377:S2CID
4338:(PDF)
4317:S2CID
4231:S2CID
4000:0.238
3977:0.136
3895:5.331
3860:3.046
3837:0.238
3814:0.136
3782:12.61
3762:12.61
2971:when
2223:delta
2219:theta
4842:PMID
4799:PMID
4668:PMID
4512:PMID
4481:2010
4439:PMID
4369:PMID
4274:PMID
4223:PMID
4188:PMID
4014:Rate
3709:and
3153:Rate
2982:>
2541:>
2530:and
2515:>
2221:and
1382:is:
935:and
878:and
820:and
766:and
673:and
82:and
35:and
4834:doi
4822:412
4789:PMC
4779:doi
4748:doi
4711:hdl
4703:doi
4691:239
4658:hdl
4650:doi
4610:hdl
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