Inverse problems for dynamical system models of cognitive processes comprise the determination of synaptic weight matrices or kernel functions for neural networks or neural/dynamic field models, respectively. We introduce dynamic cognitive modeling as a three tier top-down approach where cognitive processes are first described as algorithms that operate on complex symbolic data structures. Second, symbolic expressions and operations are represented by states and transformations in abstract vector spaces. Third, prescribed trajectories through representation space are implemented in neurodynamical systems. We discuss the Amari equation for a neural/dynamic field theory as a special case and show that the kernel construction problem is particularly ill-posed. We suggest a Tikhonov–Hebbian learning method as regularization technique and demonstrate its validity and robustness for basic examples of cognitive computations.
REFERENCES
1.
J. A. S.
Kelso
, Dynamic Patterns: The Self-Organization of Brain and Behavior
(MIT Press
, Cambridge, MA
, 1995
).2.
3.
4.
5.
Parallel Distributed Processing: Explorations in the Microstructure of Cognition
, edited by D. E.
Rumelhart
, J. L.
McClelland
, and the PDP Research Group
(MIT Press
, Cambridge, MA
, 1986
), Vol. 1
.6.
J.
Hertz
, A.
Krogh
, and R. G.
Palmer
, Introduction to the Theory of Neural Computation
(Perseus Books
, Cambridge, MA
, 1991
).7.
D.
Marr
and T.
Poggio
, Massachusetts Institute of Technology
Technical Report No. AIM-357, 2003
.8.
Z. W.
Pylyshyn
, Computation and Cognition: Toward a Foundation for Cognitive Science
(MIT Press
, Cambridge, MA
, 1986
).9.
10.
11.
P.
Smolensky
and G.
Legendre
, The Harmonic Mind: From Neural Computation to Optimality-Theoretic Grammar
,(MIT Press
, Cambridge, MA
, 2006
), Vol. 1
.12.
13.
P.
beim Graben
, S.
Gerth
, and S.
Vasishth
, Cognit. Neurodynamics
2
, 229
(2008
).14.
P.
beim Graben
, D.
Pinotsis
, D.
Saddy
, and R.
Potthast
, Cognit. Neurodynamics
2
, 79
(2008
).15.
16.
17.
R.
Haag
, Local Quantum Physics: Fields, Particles, Algebras
(Springer
, Berlin
, 1992
).18.
S. -I.
Amari
, Biol. Cybern.
27
, 77
(1977
).19.
Lectures in Supercomputational Neuroscience: Dynamics in Complex Brain Networks
, edited by P.
beim Graben
, C.
Zhou
, M.
Thiel
, and J.
Kurths
(Springer
, Berlin
, 2008
), pp. 3
–48
.20.
P. C.
Bressloff
and J. D.
Cowan
, Physica D
173
, 226
(2002
).21.
S.
Coombes
, G.
Lord
, and M.
Owen
, Physica D
178
, 219
(2003
).22.
G. B.
Ermentrout
and J. B.
McLeod
, Proc. - R. Soc. Edinburgh, Sect. A: Math
123A
, 461
(1993
).23.
24.
A.
Hutt
and F. M.
Atay
, Physica D
203
, 30
(2005
).25.
V. K.
Jirsa
and H.
Haken
, Phys. Rev. Lett.
77
, 960
(1996
).26.
P. L.
Nunez
, Behav. Brain Sci.
23
, 371
(2000
).27.
K. A.
Richardson
, S. J.
Schiff
, and B. J.
Gluckman
, Phys. Rev. Lett.
94
, 028103
(2005
).28.
P. A.
Robinson
, C. J.
Rennie
, and J. J.
Wright
, Phys. Rev. E
56
, 826
(1997
).29.
N.
Venkov
, S.
Coombes
, and P.
Matthews
, Physica D
232
, 1
(2007
).30.
H. R.
Wilson
and J. D.
Cowan
, Kybernetik
13
, 55
(1973
).31.
J. J.
Wright
, C. J.
Rennie
, G. J.
Lees
, P. A.
Robinson
, P. D.
Bourke
, C. L.
Chapman
, E.
Gordon
, and D. L.
Rowe
, Neuropsychopharmacology
28
, S80
(2003
).32.
33.
34.
E.
Thelen
, G.
Schöner
, C.
Scheier
, and L. B.
Smith
, Behav. Brain Sci.
24
, 1
(2001
).35.
A.
Newell
and H. A.
Simon
, Commun. ACM
19
, 113
(1976
).36.
J.
Fodor
and Z. W.
Pylyshyn
, Cognition
28
, 3
(1988
).37.
J. E.
Hopcroft
and J. D.
Ullman
, Introduction to Automata Theory, Languages, and Computation
(Addison-Wesley
, Menlo Park, California
, 1979
).38.
J. R.
Anderson
, Cognitive Psychology and its Implications
(W. H. Freeman
, New York, NY
, 1995
).39.
40.
A. K.
Joshi
and Y.
Schabes
, in Handbook of Formal Languages and Automata
, edited by A.
Salomma
and G.
Rosenberg
(Springer
, Berlin
, 1997
), Vol. 3
, pp. 69
–124
.41.
E. P.
Stabler
, in Logical Aspects of Comutational Linguistics
, edited by C.
Retoré
(Springer
, New York
, 1997
), pp. 68
–95
.42.
J.
Michaelis
and C.
Wartena
, in Constraints and Resources in Natural Language Syntax and Semantics
, edited by G.
Bouma
, G. -J.
Kruijff
, E.
Hinrichs
, and R. T.
Oehrle
(CSLI
, Stanford, CA
, 1999
), pp. 263
–279
.43.
44.
J.
Michaelis
, in Logical Aspects of Computational Linguistics
, edited by M.
Moortgat
(Springer
, Berlin
, 2001
), pp. 179
–198
.45.
46.
C.
Moore
, Phys. Rev. Lett.
64
, 2354
(1990
).47.
48.
R.
Badii
and A.
Politi
, Complexity: Hierarchical Structures and Scaling in Physics
(Cambridge University Press
, Cambridge
, 1997
).49.
E.
Mizraji
, Bull. Math. Biol.
51
, 195
(1989
).50.
51.
H. T.
Siegelmann
and E. D.
Sontag
, J. Comput. Syst. Sci.
50
, 132
(1995
).52.
W.
Tabor
, Technical Report No. TR98-1694, 1998
.53.
54.
W.
Tabor
, University of Connecticut Report, 2002
.55.
56.
C.
Moore
, Theor. Comput. Sci.
201
, 99
(1998
).57.
C.
Moore
and J. P.
Crutchfield
, Theor. Comput. Sci.
237
, 275
(2000
).58.
P.
beim Graben
, B.
Jurish
, D.
Saddy
, and S.
Frisch
, Int. J. Bifurcation Chaos Appl. Sci. Eng.
14
, 599
(2004
).59.
P.
beim Graben
, in Advances in Cognitive Neurodynamics
, Proceedings of the International Conference on Cognitive Neurodynamics, ICCN
2007
, edited by R.
Wang
, F.
Gu
, and E.
Shen
(Springer
, Berlin
, 2008
), pp. 469
– 473
.60.
P.
Cvitanović
, G. H.
Gunaratne
, and I.
Procaccia
, Phys. Rev. A
38
, 1503
(1988
).61.
M. B.
Kennel
and M.
Buhl
, Phys. Rev. Lett.
91
, 084102
(2003
).62.
63.
64.
65.
66.
A. R.
Edmonds
, Angular Momentum in Quantum Mechanics
(Princeton University Press
, New Jersey
, 1957
).67.
68.
69.
70.
71.
V. S.
Afraimovich
, V. P.
Zhigulin
, and M. I.
Rabinovich
, Chaos
14
, 1123
(2004
).72.
M. I.
Rabinovich
, R.
Huerta
, P.
Varona
, and V. S.
Afraimovich
, PLOS Comput. Biol.
4
, e1000072
(2008
).73.
M.
Rabinovich
, A.
Volkovskii
, P.
Lecanda
, R.
Huerta
, H. D. I.
Abarbanel
, and G.
Laurent
, Phys. Rev. Lett.
87
, 068102
(2001
).74.
R.
Huerta
and M.
Rabinovich
, Phys. Rev. Lett.
93
, 238104
(2004
).75.
76.
R.
Dale
and M. J.
Spivey
, J. Exp. Theor. Artif. Intell.
17
, 317
(2005
).77.
P.
beim Graben
and H.
Atmanspacher
, Found. Phys.
36
, 291
(2006
).78.
M. J.
Spivey
and S. E.
Anderson
, J. Exp. Theor. Artif. Intell.
20
, 239
(2008
).79.
P.
Smolensky
, in Meaning in Mind: Fodor and His Critics
, edited by B.
Loewer
and G.
Rey
(Blackwell
, Oxford
, 1991
), pp. 201
–227
.80.
81.
D. J.
Chalmers
, in Proceedings of the 12th Annual Conference on Cognitive Science Society
, 1990
(unpublished), pp. 340
–347
.82.
83.
84.
R. B.
Stein
, K. V.
Leung
, D.
Mangeron
, and M. N.
Oğuztöreli
, Kybernetik
15
, 1
(1974
).85.
H. R.
Wilson
and J. D.
Cowan
, Biophys. J.
12
, 1
(1972
).86.
A.
Arenas
, A.
Díaz-Guilera
, J.
Kurths
, Y.
Moreno
, and C.
Zhou
, Phys. Rep.
469
, 93
(2008
).87.
Lectures in Supercomputational Neuroscience: Dynamics in Complex Brain Networks
, edited by P.
beim Graben
, C.
Zhou
, M.
Thiel
, and J.
Kurths
(Springer
, Berlin
, 2008
), pp. 195
–223
.88.
89.
Neurocomputing: Foundations of Research
, edited by J. A.
Anderson
and E.
Rosenfeld
(MIT Press
, Cambridge, MA
, 1988
), Vol. 1
.90.
Neurocomputing: Directions for Research
, edited by J. A.
Anderson
, A.
Pellionisz
, and E.
Rosenfeld
(MIT Press
, Cambridge, MA
, 1990
), Vol. 2
.91.
P. S.
Churchland
and T. J.
Sejnowski
, The Computational Brain
(MIT Press
, Cambridge, MA
, 1994
).92.
The Handbook of Brain Theory and Neural Networks
, edited by M. A.
Arbib
(MIT Press
, Cambridge, MA
, 1998
).93.
C. -H.
Chen
and V.
Honavar
, IEEE Trans. Neural Netw.
10
, 1239
(1999
).94.
95.
J. L.
Elman
, in Mind as Motion: Explorations in the Dynamics of Cognition
, edited by R. F.
Port
and T.
van Gelder
(MIT Press
, Cambridge, MA
, 1995
), pp. 19
5–223
.96.
W.
Tabor
, C.
Juliano
, and M. K.
Tanenhaus
, Lang. Cognit. Processes
12
, 211
(1997
).97.
98.
S.
Lawrence
, C. L.
Giles
, and S.
Fong
, IEEE Trans. Knowl. Data Eng.
12
, 126
(2000
).99.
100.
101.
J. J.
Hopfield
, Proc. Natl. Acad. Sci. U.S.A.
79
, 2554
(1982
).102.
P.
Smolensky
, in Parallel Distributed Processing: Explorations in the Microstructure of Cognition
, edited by D. E.
Rumelhart
, J. L.
McClelland
, and the PDP Research Group (MIT Press
, Cambridge, MA
, 1986
), Vol. 1
, pp. 194
–281
.103.
H.
Haken
, Synergetic Computers and Cognition: A Top-Down Approach to Neural Nets
(Springer
, Berlin
, 1991
).104.
105.
106.
C.
Balkenius
and P.
Gärdenfors
, in Principles of Knowledge Representation and Reasoning
, edited by J. A.
Allan
, R.
Fikes
, and E.
Sandewall
(Morgan Kaufmann
, San Mateo, CA
, 1991
), pp. 32
–39
.107.
P.
Gärdenfors
, Foundations of Computation
, edited by J. v.
Eijck
and A.
Visser
(MIT-Press
, Cambridge, MA
, 1994
), pp. 49
–77
.108.
109.
E.
Mizraji
and J.
Lin
, Int. J. Bifurcation Chaos Appl. Sci. Eng.
11
, 155
(2001
).110.
111.
112.
R.
Albert
and A. -L.
Barabási
, Rev. Mod. Phys.
74
, 47
(2002
).113.
L.
Zemanová
, C.
Zhou
, and J.
Kurths
, Physica D
224
, 202
(2006
).114.
C.
Zhou
, L.
Zemanová
, G.
Zamora
, C. C.
Hilgetag
, and J.
Kurths
, Phys. Rev. Lett.
97
, 238103
(2006
).115.
116.
O.
Sporns
, D. R.
Chialvo
, M.
Kaiser
, and C. C.
Hilgetag
, Trends Cogn. Sci.
8
, 418
(2004
).117.
W.
Maass
, T.
Natschläger
, and H.
Markram
, Neural Comput.
14
, 2531
(2002
).118.
V. K.
Jirsa
and H.
Haken
, Physica D
99
, 503
(1997
).119.
V. K.
Jirsa
, A.
Fuchs
, and J. A. S.
Kelso
, Neural Comput.
10
, 2019
(1998
).120.
P. C.
Bressloff
, J. D.
Cowan
, M.
Golubitsky
, P. J.
Thomas
, and M. C.
Wiener
, Philos. Trans. R. Soc. London, Ser. B
356
, 299
(2001
).121.
P. C.
Bressloff
, J. D.
Cowan
, M.
Golubitsky
, P. J.
Thomas
, and M. C.
Wiener
, Neural Comput.
14
, 473
(2002
).122.
P. C.
Bressloff
, Phys. Rev. Lett.
89
, 088101
(2002
).123.
P. C.
Bressloff
and J. D.
Cowan
, J. Physiol. (Paris)
97
, 221
(2003
).124.
P. C.
Bressloff
, Biol. Cybern.
93
, 256
(2005
).125.
A. M.
Oster
and P. C.
Bressloff
, Bull. Math. Biol.
68
, 73
(2006
).126.
R.
Kress
, Linear Integral Equations
(Springer-Verlag
, Berlin
, 1989
).127.
D.
Colton
and R.
Kress
, Inverse Acoustic and Electromagnetic Scattering Theory
(Springer-Verlag
, Berlin
, 1998
).128.
H. W.
Engl
, M.
Hanke
, and A.
Neubauer
, Regularization of Inverse Problems
(Kluwer Academic
, Dordrecht
, 1996
).129.
A.
Neumaier
, SIAM Rev.
40
, 636
(1998
).130.
R.
Kress
, Numerical Analysis, Graduate Texts in Mathematics
(Springer-Verlag
, New York
, 1999
).131.
R.
Potthast
and P.
beim Graben
, “Existence and properties of solutions for neural field equations
,” Math. Models Meth. Appl. Sci.
(to be published).132.
R.
Potthast
and P.
beim Graben
, “Inverse problems in neural field theory
,” SIAM J. Appl. Dyn. Syst.
(unpublished).133.
H.
Bersini
, M.
Saerens
, and L. G.
Sotelino
, IEEE Trans. Neural Netw.
5
, 945
(1994
).134.
135.
B. A.
Pearlmutter
, IEEE Trans. Neural Netw.
6
, 1212
(1995
).136.
L. G.
Sotelino
, M.
Saerens
, and H.
Bersini
, Neural Netw.
7
, 767
(1994
).137.
G. -Z.
Sun
, H. -H.
Chen
, and Y. -C.
Le
, Proceedings International Joint Conference on Neural Networks (IJCNN 91)
, 1991
(unpublished), Vol. 2
, pp. 13
–18
.138.
139.
140.
M.
Breakspear
, J. A.
Roberts
, J. R.
Terry
, S.
Rodrigues
, N.
Mahant
, and P. A.
Robinson
, Cereb. Cortex
16
, 1296
(2006
).141.
Z. P.
Kilpatrick
, S. E.
Folias
, and P. C.
Bressloff
, SIAM J. Appl. Dyn. Syst.
7
, 161
(2008
).142.
V. K.
Jirsa
and J. A. S.
Kelso
, Phys. Rev. E
62
, 8462
(2000
).143.
C. J.
Rennie
, P. A.
Robinson
, and J. J.
Wright
, Biol. Cybern.
86
, 457
(2002
).144.
D.
Lehmann
, Electroencephalogr. Clin. Neurophysiol.
31
, 439
(1971
).145.
D.
Lehmann
, H.
Ozaki
, and I.
Pal
, Electroencephalogr. Clin. Neurophysiol.
67
, 271
(1987
).146.
R.
Friedrich
, A.
Fuchs
, and H.
Haken
, in Rhythms in Physiological Systems
, edited by H.
Haken
and H. P.
Koepchen
(Springer
, Berlin
, 1991
), pp. 315
–338
.147.
C.
Allefeld
and J.
Kurths
, Int. J. Bifurcation Chaos Appl. Sci. Eng.
14
, 417
(2004
).148.
149.
P.
beim Graben
, S.
Frisch
, A.
Fink
, D.
Saddy
, and J.
Kurths
, Phys. Rev. E
72
, 051916
(2005
).150.
V. K.
Jirsa
, R.
Friedrich
, and H.
Haken
, Physica D
89
, 100
(1995
).151.
H.
Obrig
and A.
Villringer
, J. Cereb. Blood Flow Metab.
23
, 1
(2003
).152.
K. J.
Friston
, P.
Fletcher
, O.
Josephs
, A.
Holmes
, M. D.
Rugg
, and R.
Turner
, Neuroimage
7
, 30
(1998
).153.
154.
O.
David
, S. J.
Kiebel
, L.
Harrison
, J.
Mattout
, J.
Kilner
, and K. J.
Friston
, Neuroimage
30
, 1255
(2006
).155.
A.
Galka
, O.
Yamashita
, T.
Ozaki
, R.
Biscay
, and P.
Valdeś-Sosa
, Neuroimage
23
, 435
(2004
).156.
A.
Galka
, T.
Ozaki
, H.
Muhle
, U.
Stephani
, and M.
Siniatchkin
, Cognit. Neurodynamics
2
, 101
(2008
).© 2009 American Institute of Physics.
2009
American Institute of Physics
You do not currently have access to this content.
