Nonlinear Functions Interrelating Neural Activity Recorded Simultaneously from Olfactory Bulb, Somatomotor, Auditory, Visual and Entorhinal Cortices of Awake, Behaving Cats

  • G. Gaál
  • W. J. Freeman


Population coding algorithms have been designed to retrodict sensory stimuli or predict motor behavior from neuronal responses (Georgopoulos et al., 1986). These include calculation of Jacobian matrices in nonlinear systems (Gaál, 1995). which made it possible to model the visuomotor hand movement task of reaching in a plane, in adaptive feedback control while updating the joint angles of a three-joint arm (Lee and Kil, 1994). The control signal was the dot product between the visual error signal and the transpose of the Jacobian matrix of the direct kinematic equation of hand movement. The trajectories of the hand were synchronized with the x and y time series outputs of coupled nonlinear equations. The equations used to calculate the adaptive feedback signal were similar to those used by Kocarev et al. (1993) to show that two different nonlinear systems could synchronize, when the difference between the goal (Lorenz system) and target signals (Chua system) was added as an adaptive feedback signal to modify the equations of the entrained (Chua) system. In robotics, the Jacobian matrix was defined by the makers of the robots. In biological systems, the matrix needs to be derived from observed time series. Experimental control and synchronization of chaos in nonlinear dynamical systems by self-controlling feedback have already been demonstrated (Pyragas, 1992; Pecora and Carroll, 1990; McKenna et al., 1994; Kelso and Ding, 1992), with applications in neurobiological control, prediction and synchronization.


Conditioned Stimulus Olfactory Bulb Jacobian Matrix Entorhinal Cortex Jacobian Matrice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. D. Bullock, S. Grossberg and F.H. Guenther, A self-organizing neural model of motor equivalent reaching and tool use by a multijoint arm, J. Cogn. Neurosci. 5 (1993) 408–435.CrossRefGoogle Scholar
  2. Y. Burnod, P. Grandguillaume, I. Otto, S. Ferraina, P.B. Johnson and R. Caminiti, Visuomotor transformations underlying arm movements toward visual targets: a neural network model of cerebral cortical operations, J. Neurosci. 12 (1992) 1435–1453.PubMedGoogle Scholar
  3. G.W. Davis and W.J. Freeman, On-One detection of respiratory events applied to behavioral conditioning in rabbits. [FEE Trans. on Biomed. Eng. 29 (1982) 453–456.Google Scholar
  4. J.N. Donoghue,.I.N. Sanes, N. Hatsopoulos, and G. Gadl, Oscillations in neural discharge and local field potentials in primate motor cortex during voluntary movement. J. Neurophys. (1996) pending.Google Scholar
  5. J.P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Modern Phys. 57 (1985) 617–656.CrossRefGoogle Scholar
  6. G. Gadl, Relationship of calculating the Jacobian matrices of nonlinear systems and population coding algorithms in neurobiology, Physics D 84, (1995) 582–600.CrossRefGoogle Scholar
  7. A.P. Georgopoulos, A. Schwartz and R.E. Kenner, Neuronal population coding of movement direction. Science 233 (1986) 1416–1419.PubMedCrossRefGoogle Scholar
  8. D. Hebb. The Organization of Behavior ( New York, Wiley. 1949 ).Google Scholar
  9. J.F. Kalaska and D.J. Crammond. Cerebral cortical mechanisms of reaching movements, Science 255 (1992) 1517–1523.PubMedCrossRefGoogle Scholar
  10. J.A.S. Kelso and M.Z. Ding, Fluctuations. intermittency and controllable chaos in motor control. In: K. Newell and D. Corcos (eds.) Variability in Motor Control. Human Kinetics. Champaign (1992).Google Scholar
  11. L. Kocarcv, A. Shang and L.O. Chus. Transitions in dynamical regimes by driving: A unified method of control and synchronization of chaos, Int. J. of Bifurcation and Chaos 3 (1993) 479–483.CrossRefGoogle Scholar
  12. A.K. Kreiter and W. Singer, Oscillatory neuronal responses in the visual cortex of the awake macaque monkey, Eur. J. Neurosci. 4: (1992) 369–375.PubMedCrossRefGoogle Scholar
  13. M. Kuperstein, Neural model of adaptive hand-eye coordination for single postures, Science. 289 (1988) 1308–1311.CrossRefGoogle Scholar
  14. S. Lee and R.M. Kil. Redundant aria kinematic control with recurrent loop, Neural Networks 7 (1994) 643–659. T.M. McKenna, T.A. McMullen and M.F. Shlesinger, The brain as a dynamic physical system. Neurosci. 60 (1994) 587–605.Google Scholar
  15. V.N. Murthy and E.E. Fetz, Coherent 25–35 Hz oscillations in the sensorimotor cortex of the awake behaving monkey, Proc. Natl. Acad. Sci. 89 (1992) 5670–5674.PubMedCrossRefGoogle Scholar
  16. L.M. Pecora and T.L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett. 64 (1990) 821–824. K. Pyragas, Continuous control of chaos by self-controlling feedback, Phys. Lett. A170 (1992) 421–428.Google Scholar
  17. J.N. Sanes and J.P. Donoghue, Oscillations in local field potentials of the primate motor cortex during voluntary movement, Proc. Natl. Acad. Sci. 90 (1993) 4470–4474.PubMedCrossRefGoogle Scholar
  18. M. Sano and Y. Sawada, Measurement of the Lyapunov spectrum from a chaotic time series, Phys. Rev. Lett. 55 (1985) 1082–1085.CrossRefGoogle Scholar
  19. A.B. Schwartz, Motor cortical activity during drawing movements: single-unit activity during sinusoidal tracing, J. Neurophys. 68 (1992) 528–541.Google Scholar
  20. A.B. Schwartz, Motor cortical activity during drawing movements: population representation during sinusoidal tracing, J. Neurophys. 70 (1993) 28–36.Google Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • G. Gaál
    • 1
  • W. J. Freeman
    • 1
  1. 1.Department of Molecular and Cell BiologyUniversity of CaliforniaBerkeleyUSA

Personalised recommendations