Symmetry Breaking in Space-Time Hierarchies Shapes Brain Dynamics and Behavior

Neuron

Volumn 94, Issue 5, 2017, Pages 1010-1026

  Pillai, Ajay S ^a,b   Jirsa, Viktor K ^c  

^a KENNEDY KRIEGER INSTITUTE   (United States)

^b JOHNS HOPKINS UNIVERSITY SCHOOL OF MEDICINE   (United States)

^c AIX MARSEILLE UNIVERSITÉ   (France)

Author keywords

behavior; brain dynamics; brain function; connectivity; dynamical system; emergence; manifold; network; symmetry; symmetry breaking

Indexed keywords

BEHAVIOR; BRAIN FUNCTION; CONNECTOME; FUNCTIONAL CONNECTIVITY; HUMAN; LOCOMOTION; MOTOR CONTROL; NERVE CELL NETWORK; NEUROANATOMY; NEUROPHYSIOLOGY; NONHUMAN; PRIORITY JOURNAL; REVIEW; SIGNAL TRANSDUCTION; BIOLOGICAL MODEL; BRAIN; NERVE TRACT; PHYSIOLOGY;

BEHAVIOR; BRAIN; HUMANS; MODELS, NEUROLOGICAL; NEURAL PATHWAYS;

EID: 85020242188     PISSN: 08966273     EISSN: 10974199     Source Type: Journal    
DOI: 10.1016/j.neuron.2017.05.013     Document Type: Review

References (144)

1. Abbott, L.F., DePasquale, B., Memmesheimer, R.-M., Building functional networks of spiking model neurons. Nat. Neurosci. 19 (2016), 350–355.
2. Almonte, F., Jirsa, V.K., Large, E.W., Tuller, B., Integration and segregation in auditory streaming. Physica D 212 (2005), 137–159.
3. Aoi, S., Yamashita, T., Tsuchiya, K., Hysteresis in the gait transition of a quadruped investigated using simple body mechanical and oscillator network models. Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 83, 2011, 061909.
4. Aoi, S., Katayama, D., Fujiki, S., Tomita, N., Funato, T., Yamashita, T., Senda, K., Tsuchiya, K., A stability-based mechanism for hysteresis in the walk-trot transition in quadruped locomotion. J. R. Soc. Interface, 10, 2013, 20120908.
5. Arnold, V.I., Ordinary Differential Equations. 1978, The MIT Press.
6. Banerjee, A., Tognoli, E., Assisi, C.G., Kelso, J.A.S., Jirsa, V.K., Mode level cognitive subtraction (MLCS) quantifies spatiotemporal reorganization in large-scale brain topographies. Neuroimage 42 (2008), 663–674.
7. Bechtel, W., Mental Mechanisms: Philosophical Perspectives on Cognitive Neuroscience. 2008, Taylor & Francis.
8. Beek, P.J., Peper, C.E., Stegeman, D.F., Dynamical models of movement coordination. Hum. Mov. Sci. 14 (1995), 573–608.
9. Bennet, M.R., Gibson, W.G., Lemon, G., Neuronal cell death, nerve growth factor and neurotrophic models: 50 years on. Auton. Neurosci. 95 (2002), 1–23.
10. Bootsma, R.J., Boulard, M., Fernandez, L., Mottet, D., Informational constraints in human precision aiming. Neurosci. Lett. 333 (2002), 141–145.
11. d'Avella, A., Saltiel, P., Bizzi, E., Combinations of muscle synergies in the construction of a natural motor behavior. Nat. Neurosci. 6 (2003), 300–308.
12. Deco, G., Jirsa, V.K., Ongoing cortical activity at rest: criticality, multistability, and ghost attractors. J. Neurosci. 32 (2012), 3366–3375.
13. Deco, G., Kringelbach, M.L., Great expectations: using whole-brain computational connectomics for understanding neuropsychiatric disorders. Neuron 84 (2014), 892–905.
14. Deco, G., Rolls, E.T., Romo, R., Stochastic dynamics as a principle of brain function. Prog. Neurobiol. 88 (2009), 1–16.
15. Deco, G., Jirsa, V., McIntosh, A.R., Sporns, O., Kötter, R., Key role of coupling, delay, and noise in resting brain fluctuations. Proc. Natl. Acad. Sci. USA 106 (2009), 10302–10307.
16. Deco, G., Jirsa, V.K., McIntosh, A.R., Emerging concepts for the dynamical organization of resting-state activity in the brain. Nat. Rev. Neurosci. 12 (2011), 43–56.
17. Deco, G., Ponce-Alvarez, A., Mantini, D., Romani, G.L., Hagmann, P., Corbetta, M., Resting-state functional connectivity emerges from structurally and dynamically shaped slow linear fluctuations. J. Neurosci. 33 (2013), 11239–11252.
18. Deco, G., Senden, M., Jirsa, V.K., How anatomy shapes dynamics: a semi-analytical study of the brain at rest by a simple spin model. Front Comput Neurosci., 6, 2012, 68.
19. Deco, G., Tononi, G., Boly, M., Kringelbach, M.L., Rethinking segregation and integration: contributions of whole-brain modelling. Nat. Rev. Neurosci. 16 (2015), 430–439.
20. Dhamala, M., Jirsa, V.K., Ding, M., Transitions to synchrony in coupled bursting neurons. Phys. Rev. Lett., 92, 2004, 028101.
21. Dusenbery, D.B., Sensory Ecology: How Organisms Acquire and Respond to Information. 1992, W. H. Freeman, New York.
22. Edelman, G.M., Neural Darwinism:The Theory of Neuronal Group Selection. 1987, Basic Books New-York.
23. Edelman, G.M., Neural Darwinism: selection and reentrant signaling in higher brain function. Neuron 10 (1993), 115–125.
24. Edelman, G.M., Gally, J.A., Degeneracy and complexity in biological systems. Proc. Natl. Acad. Sci. USA 98 (2001), 13763–13768.
25. Eliasmith, C., Stewart, T.C., Choo, X., Bekolay, T., DeWolf, T., Tang, Y., Rasmussen, D., A large-scale model of the functioning brain. Science 338 (2012), 1202–1205.
26. Fautrelle, L., Berret, B., Chiovetto, E., Pozzo, T., Bonnetblanc, F., Equilibrium constraints do not affect the timing of muscular synergies during the initiation of a whole body reaching movement. Exp. Brain Res. 203 (2010), 147–158.
27. Faye, G., and Scheel, A. (2016). Center manifolds without a phase space. arXiv Preprint arXiv:1611.07487.
28. Feldman, A.G., Once more on the equilibrium-point hypothesis (λ model) for motor control. J. Mot. Behav. 18 (1986), 17–54.
29. Feldman, A.G., Levin, M.F., The equilibrium-point hypothesis–past, present and future. Adv. Exp. Med. Biol. 629 (2009), 699–726.
30. Fields, R.D., White matter in learning, cognition and psychiatric disorders. Trends Neurosci. 31 (2008), 361–370.
31. Fink, P.W., Foo, P., Jirsa, V.K., Kelso, J.A.S., Local and global stabilization of coordination by sensory information. Exp. Brain Res. 134 (2000), 9–20.
32. Fink, P.W., Kelso, J.A.S., Jirsa, V.K., de Guzman, G., Recruitment of degrees of freedom stabilizes coordination. J. Exp. Psychol. Hum. Percept. Perform. 26 (2000), 671–692.
33. Fitts, P.M., The information capacity of the human motor system in controlling the amplitude of movement. J. Exp. Psychol. 47 (1954), 381–391.
34. Fitts, P.M., Peterson, J.R., Information capacity of discrete motor responses. J. Exp. Psychol. 67 (1964), 103–112.
35. Fornito, A., Zalesky, A., Breakspear, M., The connectomics of brain disorders. Nat. Rev. Neurosci. 16 (2015), 159–172.
36. Freeman, W.J., Pattern Recognition and Associative Memory as Dynamical Processes in a Synergetic System I, II. Biol. Cybern. 60 (1988), 17–22.
37. Friedrich, R., Jirsa, V.K., Haken, H., Uhl, C., Analyzing spatio-temporal patterns of complex systems. Nonlinear Analysis of Physiological Data, 1998, Springer, 101–114.
38. Friston, K.J., Functional and effective connectivity in neuroimaging: A synthesis. Hum. Brain Mapp. 2 (1994), 56–78.
39. Friston, K.J., Transients, metastability, and neuronal dynamics. Neuroimage 5 (1997), 164–171.
40. Friston, K., The free-energy principle: a unified brain theory?. Nat. Rev. Neurosci. 11 (2010), 127–138.
41. Friston, K., What is optimal about motor control?. Neuron 72 (2011), 488–498.
42. Friston, K.J., Daunizeau, J., Kilner, J., Kiebel, S.J., Action and behavior: a free-energy formulation. Biol. Cybern. 102 (2010), 227–260.
43. Friston, K., Mattout, J., Kilner, J., Action understanding and active inference. Biol. Cybern. 104 (2011), 137–160.
44. Friston, K.J., Kahan, J., Biswal, B., Razi, A., A DCM for resting state fMRI. Neuroimage 94 (2014), 396–407.
45. Fuchs, A., Jirsa, V.K., Kelso, J.A.S., Theory of the relation between human brain activity (MEG) and hand movements. Neuroimage 11 (2000), 359–369.
46. Ghosh, A., Rho, Y., McIntosh, A.R., Kötter, R., Jirsa, V.K., Noise during rest enables the exploration of the brain's dynamic repertoire. PLoS Comput. Biol., 4, 2008, e1000196.
47. Gold, J.I., Shadlen, M.N., The neural basis of decision making. Annu. Rev. Neurosci. 30 (2007), 535–574.
48. Goldstein, H. Jr., Poole, C.P., Safko, J.L., Classical Mechanics. 2001, Pearson.
49. Golos, M., Jirsa, V., Daucé, E., Multistability in Large Scale Models of Brain Activity. PLoS Comp Biol, 11, 2015, e1004644.
50. Golubitsky, M., Stewart, I., The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space. 2002, Birkhäuser, Basel, Boston.
51. Graziano, M.S.A., Aflalo, T.N.S., Cooke, D.F., Arm movements evoked by electrical stimulation in the motor cortex of monkeys. J. Neurophysiol. 94 (2005), 4209–4223.
52. Haken, H., Synergetics, an Introduction: Nonequilibrium Phase Transitions and Self-Organization in Physics, Chemistry, and Biology. 1983, Springer-Verlag, New York.
53. Haken, H., Principles of Brain Functioning: A Synergetic Approach to Brain Activity, Behavior and Cognition. 1996, Springer Verlag, Berlin.
54. Haken, H., Kelso, J.A., Bunz, H., A theoretical model of phase transitions in human hand movements. Biol. Cybern. 51 (1985), 347–356.
55. Hansen, E.C.A., Battaglia, D., Spiegler, A., Deco, G., Jirsa, V.K., Functional connectivity dynamics: modeling the switching behavior of the resting state. Neuroimage 105 (2015), 525–535.
56. Hebb, D.O., The organization ofbehavior. 1949, Wiley, New York.
57. Heekeren, H.R., Marrett, S., Ungerleider, L.G., The neural systems that mediate human perceptual decision making. Nat. Rev. Neurosci. 9 (2008), 467–479.
58. Holmes, P., Full, R.J., Koditschek, D., Guckenheimer, J., The Dynamics of Legged Locomotion: Models, Analyses, and Challenges. SIAM Rev. 48 (2006), 207–304.
59. Hoppensteadt, F.C., Izhikevich, E.M., Weakly Connected Neural Networks. 1997, Springer.
60. Horwitz, B., Tagamets, M.A., McIntosh, A.R., Neural modeling, functional brain imaging, and cognition. Trends Cogn. Sci. 3 (1999), 91–98.
61. Hutchins, J.B., Barger, S.W., Why neurons die: cell death in the nervous system. Anat. Rec. 253 (1998), 79–90.
62. Hutchison, R.M., Womelsdorf, T., Allen, E.A., Bandettini, P.A., Calhoun, V.D., Corbetta, M., Della Penna, S., Duyn, J.H., Glover, G.H., Gonzalez-Castillo, J., et al. Dynamic functional connectivity: promise, issues, and interpretations. Neuroimage 80 (2013), 360–378.
63. Huys, R., Smeeton, N.J., Hodges, N.J., Beek, P.J., Williams, A.M., On the dynamic information underlying visual anticipation skill. Percept. Psychophys. 70 (2008), 1217–1234.
64. Huys, R., Studenka, B.E., Rheaume, N.L., Zelaznik, H.N., Jirsa, V.K., Distinct timing mechanisms produce discrete and continuous movements. PLoS Comput. Biol., 4, 2008, e1000061.
65. Huys, R., Fernandez, L., Bootsma, R.J., Jirsa, V.K., Fitts’ law is not continuous in reciprocal aiming. Proc. Biol. Sci. 277 (2010), 1179–1184.
66. Huys, R., Perdikis, D., Jirsa, V.K., Functional architectures and structured flows on manifolds: a dynamical framework for motor behavior. Psychol. Rev. 121 (2014), 302–336.
67. Hwang, E.J., Shadmehr, R., Internal models of limb dynamics and the encoding of limb state. J. Neural Eng. 2 (2005), S266–S278.
68. Jaeger, H., Haas, H., Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication. Science 304 (2004), 78–80.
69. Jerbi, K., Lachaux, J.-P., N'Diaye, K., Pantazis, D., Leahy, R.M., Garnero, L., Baillet, S., Coherent neural representation of hand speed in humans revealed by MEG imaging. Proc. Natl. Acad. Sci. USA 104 (2007), 7676–7681.
70. Jirsa, V.K., Neural field dynamics with local and global connectivity and time delay. Philos Trans A Math Phys Eng Sci 367 (2009), 1131–1143.
71. Jirsa, V.K., Kelso, J.A.S., Spatiotemporal pattern formation in neural systems with heterogeneous connection topologies. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62:6 Pt B (2000), 8462–8465.
72. Jirsa, V.K., Kelso, J.A.S., The excitator as a minimal model for the coordination dynamics of discrete and rhythmic movement generation. J. Mot. Behav. 37 (2005), 35–51.
73. Jirsa, V.K., McIntosh, A.R., Handbook of Brain Connectivity. 2007, Springer.
74. Jirsa, V.K., Fuchs, A., Kelso, J.A.S., Connecting cortical and behavioral dynamics: bimanual coordination. Neural Comput. 10 (1998), 2019–2045.
75. Jonas, E., Kording, K.P., Could a neuroscientist understand a microprocessor?. PLoS Comput. Biol., 13, 2017, e1005268.
76. Kelso, J.A.S., Dynamic Patterns: The Self-Organization of Brain and Behavior. 1995, MIT Press.
77. Kelso, J.A.S., Multistability and metastability: understanding dynamic coordination in the brain. Philos. Trans. R. Soc. Lond. B Biol. Sci. 367 (2012), 906–918.
78. Kelso, J.A., Tuller, B., Converging evidence in support of common dynamical principles for speech and movement coordination. Am. J. Physiol. 246 (1984), R928–R935.
79. Kelso, J.A.S., Fuchs, A., Lancaster, R., Holroyd, T., Cheyne, D., Weinberg, H., Dynamic cortical activity in the human brain reveals motor equivalence. Nature 392 (1998), 814–818.
80. Kiebel, S.J., Daunizeau, J., Friston, K.J., Perception and hierarchical dynamics. Front. Neuroinform., 3, 2009, 20.
81. Krakauer, J.W., Ghazanfar, A.A., Gomez-Marin, A., MacIver, M.A., Poeppel, D., Neuroscience Needs Behavior: Correcting a Reductionist Bias. Neuron 93 (2017), 480–490.
82. Krüger, M., Borbély, B., Eggert, T., Straube, A., Synergistic control of joint angle variability: influence of target shape. Hum. Mov. Sci. 31 (2012), 1071–1089.
83. Kugler, P.N., Kelso, J.A.S., Turvey, M.T., On the concept of coordinative structures as dissipative structures: I. Theoretical lines of convergence. Tutorials in Motor Behavior, 1980, 3–47.
84. Kuramoto, Y., Chemical Oscillations, Waves, and Turbulence. 2003, Dover Publications.
85. Lamoth, C.J.C., Daffertshofer, A., Huys, R., Beek, P.J., Steady and transient coordination structures of walking and running. Hum. Mov. Sci. 28 (2009), 371–386.
86. Lashley, K.S., Basic neural mechanisms in behavior. Psychol. Rev. 37 (1930), 1–24.
87. Latash, M.L., The bliss (not the problem) of motor abundance (not redundancy). Exp. Brain Res. 217 (2012), 1–5.
88. Levitis, D.A., Lidicker, W.Z. Jr., Freund, G., Behavioural biologists don't agree on what constitutes behaviour. Anim. Behav. 78 (2009), 103–110.
89. Lukoševičius, M., Jaeger, H., Schrauwen, B., Reservoir Computing Trends. KI-Künstliche Intelligenz 26 (2012), 365–371.
90. Maass, W., Natschläger, T., Markram, H., Real-time computing without stable states: a new framework for neural computation based on perturbations. Neural Comput. 14 (2002), 2531–2560.
91. Maass, W., Joshi, P., Sontag, E.D., Computational aspects of feedback in neural circuits. PLoS Comput. Biol., 3, 2007, e165.
92. Makeig, S., Westerfield, M., Jung, T.-P., Enghoff, S., Townsend, J., Courchesne, E., Sejnowski, T.J., Dynamic brain sources of visual evoked responses. Science 295 (2002), 690–694.
93. Marder, E., Goaillard, J.-M., Variability, compensation and homeostasis in neuron and network function. Nat. Rev. Neurosci. 7 (2006), 563–574.
94. Mattos, D.J.S., Latash, M.L., Park, E., Kuhl, J., Scholz, J.P., Unpredictable elbow joint perturbation during reaching results in multijoint motor equivalence. J. Neurophysiol. 106 (2011), 1424–1436.
95. McIntyre, J., Bizzi, E., Servo Hypotheses for the Biological Control of Movement. J. Mot. Behav. 25 (1993), 193–202.
96. Meyer-Lindenberg, A., Ziemann, U., Hajak, G., Cohen, L., Berman, K.F., Transitions between dynamical states of differing stability in the human brain. Proc. Natl. Acad. Sci. USA 99 (2002), 10948–10953.
97. Moran, D.W., Schwartz, A.B., Motor Cortical Representation of Speed and Direction During Reaching. Journal of Neurophysiology 82 (1999), 2676–2692.
98. Mottet, D., Bootsma, R.J., The dynamics of goal-directed rhythmical aiming. Biol. Cybern. 80 (1999), 235–245.
99. Mottet, D., Guiard, Y., Ferrand, T., Bootsma, R.J., Two-handed performance of a rhythmical fitts task by individuals and dyads. J. Exp. Psychol. Hum. Percept. Perform. 27 (2001), 1275–1286.
100. Nicolis, G., Prigogine, I., Self-Organization in Non-Equilibrium Systems: from dissipative structures to order through fluctuations. 1977, Wiley.
101. Nowotny, T., Rabinovich, M.I., Dynamical origin of independent spiking and bursting activity in neural microcircuits. Phys. Rev. Lett., 98, 2007, 128106.
102. Park, J., Zatsiorsky, V.M., Latash, M.L., Optimality vs. variability: an example of multi-finger redundant tasks. Exp. Brain Res. 207 (2010), 119–132.
103. Park, J., Singh, T., Zatsiorsky, V.M., Latash, M.L., Optimality versus variability: effect of fatigue in multi-finger redundant tasks. Exp. Brain Res. 216 (2012), 591–607.
104. Perdikis, D., Huys, R., Jirsa, V., Complex processes from dynamical architectures with time-scale hierarchy. PLoS ONE, 6, 2011, e16589.
105. Perko, L., Differential Equations and Dynamical Systems. 2006, Springer.
106. Petkoski, S., Spiegler, A., Proix, T., Aram, P., Temprado, J.-J., Jirsa, V.K., Heterogeneity of time delays determines synchronization of coupled oscillators. Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 94, 2016, 012209.
107. Rabinovich, M.I., Varona, P., Selverston, A.I., Abarbanel, H.D.I., Dynamical principles in neuroscience. Rev. Mod. Phys. 78 (2006), 1213–1265.
108. Raichle, M.E., The restless brain. Brain Connect. 1 (2011), 3–12.
109. Razi, A., Kahan, J., Rees, G., Friston, K.J., Construct validation of a DCM for resting state fMRI. Neuroimage 106 (2015), 1–14.
110. Robinson, P.A., Patchy propagators, brain dynamics, and the generation of spatially structured gamma oscillations. Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 73, 2006, 041904.
111. Rubin, J.J., Rubin, J.E., Ermentrout, G.B., Analysis of synchronization in a slowly changing environment: how slow coupling becomes fast weak coupling. Phys. Rev. Lett., 110, 2013, 204101.
112. Sanz Leon, P., Knock, S.A., Woodman, M.M., Domide, L., Mersmann, J., McIntosh, A.R., Jirsa, V., The Virtual Brain: a simulator of primate brain network dynamics. Front. Neuroinform., 7, 2013, 10.
113. Sanz-Leon, P., Knock, S.A., Spiegler, A., Jirsa, V.K., Mathematical framework for large-scale brain network modeling in The Virtual Brain. Neuroimage 111 (2015), 385–430.
114. Scholz, J.P., Schöner, G., The uncontrolled manifold concept: identifying control variables for a functional task. Exp. Brain Res. 126 (1999), 289–306.
115. Schöner, G., Recent Developments and Problems in Human Movement Science and Their Conceptual Implications. Ecol. Psychol. 7 (1995), 291–314.
116. Sengupta, B., Tozzi, A., Cooray, G.K., Douglas, P.K., Friston, K.J., Towards a neuronal gauge theory. PLoS Biol., 14, 2016, e1002400.
117. Shannon, C.E., Weaver, W., The mathematical theory of communication. 1949, The University of Illinois Press.
118. Shevchenko, M., Windridge, D., Kittler, J., A linear-complexity reparameterisation strategy for the hierarchical bootstrapping of capabilities within perception–action architectures. Image Vis. Comput. 27 (2009), 1702–1714.
119. Spiegler, A., Hansen, E., Jirsa, V.K., Spatiotemporal dynamics in the human brain during rest:a virtual brain study. BMC Neurosci., 14, 2013, 195.
120. Spiegler, A., Hansen, E.C., Bernard, C., McIntosh, A.R., Jirsa, V.K., Selective Activation of Resting-State Networks following Focal Stimulation in a Connectome-Based Network Model of the Human Brain. eNeuro, 3, 2016 ENEURO.0068-16.2016.
121. Sporns, O., Networks of the Brain. 2010, The MIT Press, Cambridge, Mass.
122. Sporns, O., Discovering the Human Connectome. 2012, The MIT Press, Cambridge, Mass.
123. Sporns, O., Tononi, G., Kötter, R., The human connectome: A structural description of the human brain. PLoS Comput. Biol., 1, 2005, e42.
124. Strogatz, S.H., Nonlinear dynamics and chaos. With applications to physics, biology, chemistry, and engineering. 1994, Perseus, Cambridge, Massachusetts.
125. Sussillo, D., Neural circuits as computational dynamical systems. Curr. Opin. Neurobiol. 25 (2014), 156–163.
126. Sussillo, D., Abbott, L.F., Generating coherent patterns of activity from chaotic neural networks. Neuron 63 (2009), 544–557.
127. Sussillo, D., Abbott, L.F., Transferring learning from external to internal weights in echo-state networks with sparse connectivity. PLoS ONE, 7, 2012, e37372.
128. Tinbergen, N., The study of instinct. 1951, Clarendon Press/Oxford University Press, New York, NY, US.
129. Tononi, G., Sporns, O., Edelman, G.M., Measures of degeneracy and redundancy in biological networks. Proc. Natl. Acad. Sci. USA 96 (1999), 3257–3262.
130. Troje, N.F., Decomposing biological motion: a framework for analysis and synthesis of human gait patterns. J. Vis. 2 (2002), 371–387.
131. Turvey, M.T., Coordination. Am. Psychol. 45 (1990), 938–953.
132. Uhlhaas, P.J., Singer, W., Neural synchrony in brain disorders: relevance for cognitive dysfunctions and pathophysiology. Neuron 52 (2006), 155–168.
133. Uhlhaas, P.J., Pipa, G., Lima, B., Melloni, L., Neuenschwander, S., Nikolić, D., Singer, W., Neural synchrony in cortical networks: history, concept and current status. Front. Integr. Nuerosci., 3, 2009, 17.
134. van der Steen, M.M.C., Bongers, R.M., Joint angle variability and co-variation in a reaching with a rod task. Exp. Brain Res. 208 (2011), 411–422.
135. van Mourik, A.M., Daffertshofer, A., Beek, P.J., Deterministic and stochastic features of rhythmic human movement. Biol. Cybern. 94 (2006), 233–244.
136. van Mourik, A.M., Daffertshofer, A., Beek, P.J., Estimating Kramers–Moyal coefficients in short and non-stationary data sets. Phys. Lett. A 351 (2006), 13–17.
137. von Holst, E., The behavioural physiology of animals and man: the collected papers of Erich von Holst. 1973, University of Miami Press, Coral Gables, Florida.
138. Wang, X.-J., Decision making in recurrent neuronal circuits. Neuron 60 (2008), 215–234.
139. Wilimzig, C., Schneider, S., Schöner, G., The time course of saccadic decision making: dynamic field theory. Neural Netw. 19 (2006), 1059–1074.
140. Wilson, H.R., Cowan, J.D., Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J. 12 (1972), 1–24.
141. Wolpert, D.M., Computational approaches to motor control. Trends Cogn. Sci. 1 (1997), 209–216.
142. Wolpert, D.M., Ghahramani, Z., Computational principles of movement neuroscience. Nat. Neurosci. 3:Suppl (2000), 1212–1217.
143. Woodworth, R.S., Accuracy of voluntary movement. The Psychological Review: Monograph Supplements, 3, 1899, i-114.
144. Yuste, R., From the neuron doctrine to neural networks. Nat. Rev. Neurosci. 16 (2015), 487–497.