Inverse problems in neural field theory
To link to this article DOI: 10.1137/080731220
We study inverse problems in neural field theory, i.e., the construction of synaptic weight kernels yielding a prescribed neural field dynamics. We address the issues of existence, uniqueness, and stability of solutions to the inverse problem for the Amari neural field equation as a special case, and prove that these problems are generally ill-posed. In order to construct solutions to the inverse problem, we first recast the Amari equation into a linear perceptron equation in an infinite-dimensional Banach or Hilbert space. In a second step, we construct sets of biorthogonal function systems allowing the approximation of synaptic weight kernels by a generalized Hebbian learning rule. Numerically, this construction is implemented by the Moore–Penrose pseudoinverse method. We demonstrate the instability of these solutions and use the Tikhonov regularization method for stabilization and to prevent numerical overfitting. We illustrate the stable construction of kernels by means of three instructive examples.
 S.-I. Amari, Dynamics of pattern formation in lateral-inhibition type neural fields, Biolog. Cybernet., 27 (1977), pp. 77–87.  J. A. Anderson and E. Rosenfeld, eds., Neurocomputing. Foundations of Research, Vol. 1, MIT Press, Cambridge, MA, 1988.  P. beim Graben, Foundations of neurophysics, in Lectures in Supercomputational Neuroscience: Dynamics in Complex Brain Networks, P. beim Graben, C. Zhou, M. Thiel, and J. Kurths, eds., Springer, Berlin, 2008, pp. 3–48.  P. beim Graben and J. Kurths, Simulating global properties of electroencephalograms with minimal random neural networks, Neurocomputing, 71 (2008), pp. 999–1007.  P. beim Graben, T. Liebscher, and J. Kurths, Neural and cognitive modeling with networks of leaky integrator units, in Lectures in Supercomputational Neuroscience: Dynamics in Complex Brain Networks, P. beim Graben, C. Zhou, M. Thiel, and J. Kurths, eds., Springer, Berlin, 2008, pp. 195–223.  P. beim Graben, D. Pinotsis, D. Saddy, and R. Potthast, Language processing with dynamic fields, Cognitive Neurodynamics, 2 (2008), pp. 79–88.  H. Bersini, M. Saerens, and L. G. Sotelino, Hopfield net generation, encoding and classification of temporal trajectories, IEEE Trans. Neural Networks, 5 (1994), pp. 945–953.  E. Boasso, On the Moore-Penrose inverse, EP Banach space operators, and EP Banach algebra elements, J. Math. Anal. Appl., 339 (2008), pp. 1003–1014.  M. Breakspear, J. A. Roberts, J. R. Terry, S. Rodrigues, N. Mahant, and P. A. Robinson, A unifying explanation of primary generalized seizures through nonlinear brain modeling and bifurcation analysis, Cerebral Cortex, 16 (2006), pp. 1296–1313.  P. C. Bressloff, Bloch waves, periodic feature maps, and cortical pattern formation, Phys. Rev. Lett., 89 (2002), paper 088101.  D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, New York, Berlin, 1998.  S. Coombes, G. J. Lord, and M. R. Owen, Waves and bumps in neuronal networks with axo-dendritic synaptic interactions, Phys. D, 178 (2003), pp. 219–241.  H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Springer, Berlin, 1996.  W. Erlhagen and G. Sch¨oner, Dynamic field theory of movement preparation, Psych. Rev., 109 (2002), pp. 545–572.  G. B. Ermentrout and J. B. McLeod, Existence and uniqueness of travelling waves for a neural network, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), pp. 461–478.  J. S. Griffith, A field theory of neural nets: I. Derivation of field equations, Bull. Math. Biophys., 25 (1963), pp. 111–120.  H. Haken, Synergetics. An Introduction, Springer, New York, 1983.  D. O. Hebb, The Organization of Behavior, Wiley, New York, 1949; partly reprinted in .  J. Hertz, A. Krogh, and R. G. Palmer, Introduction to the Theory of Neural Computation, Perseus Books, Cambridge, MA, 1991.  J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Natl. Acad. Sci. USA, 79 (1982), pp. 2554–2558.  A. Hutt and F. M. Atay, Analysis of nonlocal neural fields for both general and gamma-distributed connectivities, Phys. D, 203 (2005), pp. 30–54.  V. K. Jirsa and H. Haken, Field theory of electromagnetic brain activity, Phys. Rev. Lett., 77 (1996), pp. 960–963.  R. Kress, Linear Integral Equations, Springer, Berlin, 1989.  R. Kress, Numerical Analysis, Springer, Berlin, 1998.  T. Liebscher, Modeling reaction times with neural networks using leaky integrator units, in Proceedings of the 18th Twente Workshop on Language Technology (TWLT 18), K. Jokinen, D. Heylen, and A. Nijholt, eds., University of Twente, Twente, The Netherlands, 2000, pp. 81–94.  M. Minsky and S. Papert, Perceptrons, MIT Press, Cambridge, MA, 1969; partly reprinted in .  B. A. Pearlmutter, Learning state space trajectories in recurrent neural networks, Neural Comput., 1 (1989), pp. 263–269.  B. A. Pearlmutter, Gradient calculations for dynamic recurrent neural networks: A survey, IEEE Trans. Neural Networks, 6 (1995), pp. 1212–1228.  R. Potthast, Point-Sources and Multipoles in Inverse Scattering Theory, Chapman & Hall, London, 2001.  R. Potthast, Topical review: A survey on sampling and probe methods for inverse problems, Inverse Problems, 22 (2006), pp. R1–R47.  R. Potthast and P. beim Graben, Existence and properties of solutions for neural field equations, Math. Methods Appl. Sci., to appear; DOI: 10.1002/mmm.1199.  K. A. Richardson, S. J. Schiff, and B. J. Gluckman, Control of traveling waves in the mammalian cortex, Phys. Rev. Lett., 94 (2005), paper 028103.  P. A. Robinson, C. J. Rennie, and J. J. Wright, Propagation and stability of waves of electrical activity in the cerebral cortex, Phys. Rev. E, 56 (1997), pp. 826–840.  F. Rosenblatt, The perceptron: A probabilistic model for information storage and organization in the brain, Phys. Rev., 65 (1958), pp. 386–408; also reprinted in .  G. Sch¨oner and E. Thelen, Using dynamic field theory to rethink infant habituation, Psych. Rev., 113 (2006), pp. 273–299.  L. G. Sotelino, M. Saerens, and H. Bersini, Classification of temporal trajectories by continuous-time recurrent nets, Neural Networks, 7 (1994), pp. 767–776.  R. B. Stein, K. V. Leung, D. Mangeron, and M. N. O˘guzt¨oreli, Improved neuronal models for studying neural networks, Kybernetik, 15 (1974), pp. 1–9.  G.-Z. Sun, H.-H. Chen, and Y.-C. Le, A fast online learning algorithm for recurrent neural networks, in Proceedings of the International Joint Conference on Neural Networks (IJCNN 91), Vol. 2, 1991, pp. 13–18.  E. Thelen, G. Sch¨oner, C. Scheier, and L. B. Smith, The dynamics of embodiment: A field theory of infant perseverative reaching, Behavioral Brain Sci., 24 (2001), pp. 1–86.  N. A. Venkov, S. Coombes, and P. C. Matthews, Dynamic instabilities in scalar neural field equations with space-dependent delays, Phys. D, 232 (2007), pp. 1–15.  H. R. Wilson and J. D. Cowan, Excitatory and inhibitory interactions in localized populations of model neurons, Biophys. J., 12 (1972), pp. 1–24.  H. R. Wilson and J. D. Cowan, A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue, Kybernetik, 13 (1973), pp. 55–80.
Centaur Editors: Update this record