Image Segmentation by Complex-Valued UnitsImage Segmentation by Complex-Valued Units

Cornelius WeberHybrid Intelligent Systems

School of Computing and TechnologyUniversity of Sunderland

Presented at the Perceptual Dynamics Laboratory, RIKEN 8th December 2005

ContentsContents

• Attractor Network which Converges

• Non-Convergence and Spike Synchrony

• Coupled Chaotic Oscillators for Spike Phases

• Outlook

ContentsContents

• Attractor Network which Converges

• Non-Convergence and Spike Synchrony

• Coupled Chaotic Oscillators for Spike Phases

• Outlook

Attractor Network:Attractor Network:Competition via RelaxationCompetition via Relaxation

Attractor Network:Attractor Network:Competition via RelaxationCompetition via Relaxation

weight profile rate profile

rate update

ri(t+1) = f (Σj wij rj(t))

winner

Response CharacteristicsResponse CharacteristicsResponse CharacteristicsResponse Characteristics

linear sparse competitive

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

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

Learning Object RecognitionLearning Object Recognition

attractor network Active units (features) not separated

Binding- and learning problem?green

redbackground

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)

Stringer, S.M. and Rolls, E.T. Position invariant recognition in the visual system with cluttered environments. Neural Networks 13, 305-15 (2000)

ContentsContents

• Attractor Network which Converges

• Non-Convergence and Spike Synchrony

• Coupled Chaotic Oscillators for Spike Phases

• Outlook

Necker CubeNecker CubeNecker CubeNecker Cube

Attractor networks that minimize an energy function do not account for bi-stability

Neuronal Spike ChaosNeuronal Spike ChaosNeuronal Spike ChaosNeuronal 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)

Banerjee, A. On the Phase-Space Dynamics of Systems of Spiking Neurons. I: Model and Experiments. Neural Computation, 13(1), 161-93 (2001)

Neuronal SynchronyNeuronal SynchronyNeuronal SynchronyNeuronal 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)

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)

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)

Neuronal Spike ChaosNeuronal Spike ChaosNeuronal Spike ChaosNeuronal 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)

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)

ContentsContents

• Attractor Network which Converges

• Non-Convergence and Spike Synchrony

• Coupled Chaotic Oscillators for Spike Phases

• Outlook

Complex NumberComplex NumberComplex NumberComplex Number

φr r rate

φ phase

z = r eiφ

zi

1

= r cos φ + i r sin φ

Deterministic ChaosDeterministic ChaosDeterministic ChaosDeterministic Chaos

Logistic map:

Ф(t+1) = A Ф(t) (1- Ф(t))

Phase φ = 2π Ф

Coupling of the PhasesCoupling of the PhasesCoupling of the PhasesCoupling of the Phases

For phases: Σj wkj rj eiφ ≡ zkwf}

coupling strength for phasescomplex number

}

“Net input” to neuron k:

For rates: Σj wkj rj

j

weighted field

Relaxation of the PhasesRelaxation of the PhasesRelaxation of the PhasesRelaxation of the Phases

Compute “net input”: zkwf = Σj wkj rj eiφ

Compute new phase: Фk(t+1) = A Фkwf(t) (1- Фk

wf(t))

(remember: φ = 2π Ф)

From zwf, take phase φwf

j

Relaxation of Rates and PhasesRelaxation of Rates and PhasesRelaxation of Rates and PhasesRelaxation of Rates and Phases

Phase of any neuron behaves chaoticallyCoupled neurons have similar phases

Phase Separation HistogramPhase Separation HistogramPhase Separation HistogramPhase Separation Histogram

Large phase differences at boundary of activation hill

Toward Learning Object RecognitionToward Learning Object Recognition

attractor network

Toward Learning Object RecognitionToward Learning Object Recognition

attractor network

ContentsContents

• Attractor Network which Converges

• Non-Convergence and Spike Synchrony

• Coupled Chaotic Oscillators for Spike Phases

• Outlook

Plans and QuestionsPlans and QuestionsPlans and QuestionsPlans 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?