1. Image Segmentation by Complex-Valued Units Cornelius Weber Hybrid Intelligent Systems School of Computing and Technology University of Sunderland Presented at the Perceptual Dynamics Laboratory, RIKEN 8 th December 2005

2. Contents Attractor Network which Converges Non-Convergence and Spike Synchrony Coupled Chaotic Oscillators for Spike Phases Outlook

3. Contents Attractor Network which Converges Non-Convergence and Spike Synchrony Coupled Chaotic Oscillators for Spike Phases Outlook

4. Attractor Network: Competition via Relaxation weight profile rate profile rate update r i (t+1) = f ( Σ j w ij r j (t) )

5. winner Response Characteristics linearsparsecompetitive Weber, C. Self-Organization of Orientation Maps, Lateral Connections, and Dynamic Receptive Fields in the Primary Visual Cortex. Proc. ICANN (2001)

6. Learning Object Recognition attractor network Active units (features) not separated Binding- and learning problem? green red background apple Learning objects in cluttered background is difficult Stringer, S.M. and Rolls, E.T. Position invariant recognition in the visual system with cluttered environments. Neural Networks 13, 305-15 (2000)

7. Contents Attractor Network which Converges Non-Convergence and Spike Synchrony Coupled Chaotic Oscillators for Spike Phases Outlook

8. Necker Cube Attractor networks that minimize an energy function do not account for bi-stability

9. Neuronal Spike Chaos A wide range of spiking neuron models displays three distinct categories of behavior: - quiescence - intense periodic seizure-like activity - sustained chaos in normal operational conditions Banerjee, A. On the Phase-Space Dynamics of Systems of Spiking Neurons. I: Model and Experiments. Neural Computation, 13(1), 161-93 (2001)

10. Neuronal Synchrony “cortical neurons often engage in oscillatory activity which is not stimulus locked but caused by internal interactions” “activity synchronization was present in the expectation period before stimulus presentation and could not be induced de novo by the stimulus” Singer, W. Synchronization, Bining and Expectancy. In: The Handbook of Brain Theory and Neural Networks, pp. 1136-43 (2003) Cardoso de Oliviera, S., Thiele, A. and Hoffmann, K.P. Synchronization of neuronal activity during stimulus expectation in a direction discrimination task. J. Neurosci., 17, 9248-60 (1997)

11. Neuronal Spike Chaos We need a method to: - create patterns of synchronization - avoid long-term stabilization (bi-stability is welcome!) van Leeuwen, C., Steyvers, M. and Nooter, M. Stability and Intermittency in Large-Scale Coupled Oscillator Models for Perceptual Segmentation. J. Mathematical Psychology, 41(4), 319-44 (1997)

12. Contents Attractor Network which Converges Non-Convergence and Spike Synchrony Coupled Chaotic Oscillators for Spike Phases Outlook

13. Complex Number φ r r rate φ phase z = r e iφ z i 1 = r cos φ + i r sin φ

14. Deterministic Chaos Logistic map: Ф(t+1) = A Ф(t) (1- Ф(t)) Phase φ = 2π Ф

15. Coupling of the Phases For phases: Σ j w kj r j e iφ ≡ z k wf } coupling strength for phases complex number } “Net input” to neuron k: For rates: Σ j w kj r j j weighted field

16. Relaxation of the Phases Compute “net input”: z k wf = Σ j w kj r j e iφ Compute new phase: Ф k (t+1) = A Ф k wf (t) (1- Ф k wf (t)) (remember: φ = 2π Ф) From z wf, take phase φ wf j

17. Relaxation of Rates and Phases Phase of any neuron behaves chaotically Coupled neurons have similar phases

18. Phase Separation Histogram Large phase differences at boundary of activation hill

19. Toward Learning Object Recognition attractor network

20. Toward Learning Object Recognition attractor network

21. Contents Attractor Network which Converges Non-Convergence and Spike Synchrony Coupled Chaotic Oscillators for Spike Phases Outlook

22. Plans and Questions - The higher hierarchical level shall benefit! - Should the rates depend on the phases? → This would influence learning! - Learning with Phase Timing Dependent Plasticity?