1. Image Segmentation by Complex-Valued Units Cornelius Weber and Stefan Wermter Hybrid Intelligent Systems School of Computing and Technology University of Sunderland ICANN Conference, September 2005

2. Project funded by the Future and Emerging Technologies arm of the IST Programme FET-Open scheme ContentsContents Attractor Network which Converges Non-Convergence and Spike Synchrony Coupled Oscillators for Spike Phases Outlook

3. Project funded by the Future and Emerging Technologies arm of the IST Programme FET-Open scheme ContentsContents Attractor Network which Converges Non-Convergence and Spike Synchrony Coupled Oscillators for Spike Phases Outlook

4. Project funded by the Future and Emerging Technologies arm of the IST Programme FET-Open scheme Attractor Network: Competition via Relaxation weight profile rate profile rate update r i (t+1) = f ( Σ ij w ij r j (t) )

5. Project funded by the Future and Emerging Technologies arm of the IST Programme FET-Open scheme 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. Project funded by the Future and Emerging Technologies arm of the IST Programme FET-Open scheme 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. Project funded by the Future and Emerging Technologies arm of the IST Programme FET-Open scheme ContentsContents Attractor Network which Converges Non-Convergence and Spike Synchrony Coupled Oscillators for Spike Phases Outlook

8. Project funded by the Future and Emerging Technologies arm of the IST Programme FET-Open scheme Neckar Cube Attractor networks that minimize an energy function do not account for bi-stability

9. Project funded by the Future and Emerging Technologies arm of the IST Programme FET-Open scheme Neuronal Spike Chaos A wide range of spiking neuron models displays three distinct categories of behaviour: - 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. Project funded by the Future and Emerging Technologies arm of the IST Programme FET-Open scheme 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. Project funded by the Future and Emerging Technologies arm of the IST Programme FET-Open scheme Neuronal Spike Chaos We need a method to: - create patterns of synchronization - avoid long-term stabilization (bi-stability is welcome!) van Leeuven, 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. Project funded by the Future and Emerging Technologies arm of the IST Programme FET-Open scheme ContentsContents Attractor Network which Converges Non-Convergence and Spike Synchrony Coupled Oscillators for Spike Phases Outlook

13. Project funded by the Future and Emerging Technologies arm of the IST Programme FET-Open scheme Complex Number φ r r rate φ phase z = r e iφ z i 1 = r cos φ + i r sin φ

14. Project funded by the Future and Emerging Technologies arm of the IST Programme FET-Open scheme Deterministic Chaos Logistic map: Ф(t+1) = A Ф(t) (1- Ф(t)) Phase φ = 2π Ф

15. Project funded by the Future and Emerging Technologies arm of the IST Programme FET-Open scheme Coupling of the Phases For phases: Σ kj w kj r j e iφ ≡ z k wf } coupling strength for phases complex number } “Net input” to neuron k: For rates: Σ kj w kj r j j weighted field

16. Project funded by the Future and Emerging Technologies arm of the IST Programme FET-Open scheme Relaxation of the Phases Compute “net input”: z k wf = Σ kj 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. Project funded by the Future and Emerging Technologies arm of the IST Programme FET-Open scheme Relaxation of Rates and Phases Phase of any neuron behaves chaotically Coupled neurons have similar phases

18. Project funded by the Future and Emerging Technologies arm of the IST Programme FET-Open scheme Phase Separation Histogram Large phase differences at boundary of activation hill

19. Project funded by the Future and Emerging Technologies arm of the IST Programme FET-Open scheme Toward Learning Object Recognition attractor network

20. Project funded by the Future and Emerging Technologies arm of the IST Programme FET-Open scheme Toward Learning Object Recognition attractor network

21. Project funded by the Future and Emerging Technologies arm of the IST Programme FET-Open scheme ContentsContents Attractor Network which Converges Non-Convergence and Spike Synchrony Coupled Oscillators for Spike Phases Outlook

22. Project funded by the Future and Emerging Technologies arm of the IST Programme FET-Open scheme 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?