Localized Excitations in Networks of Spiking Neurons Hecke Schrobsdorff Bernstein Center for Computational Neuroscience Göttingen Max Planck Institute for Dynamics and Self-Organization Seminar: Irreversible Prozesse und Selbstorganisation Berlin 24.01.06
Outline 1 Introduction 2 Localized Excitations 3 Existence 4 Simulation 5 Feature Binding Model
Introduction Source of Inspiration: Discrete Breathers Discrete breathers Lattice of weakly coupled oscillators Localized persistent modes are called breathers Energy trap Localized excitations in neural networks have strong analogies, but: Delta-like interactions via spikes Non-oscillatory units (integrate-and-fire neurons) S. Aubry (1997) Physica D S. Flach, C.R. Willis (1998) Physics Reports
Localized Excitations Single Unit Model v vth v vreset t Leaky integrate n fire neurons: resting potential v threshold potential v th refraction potential v reset
Localized Excitations Network Structure spatial organization in a chain delta coupling equal synaptic weights ǫ only to the two nearest neighbours, all excitatory synaptic delay τ no synaptic changes, i.e. no learning network size arbitrary, only localized activations are of interest
Localized Excitations Network Structure spatial organization in a chain delta coupling equal synaptic weights ǫ only to the two nearest neighbours, all excitatory synaptic delay τ no synaptic changes, i.e. no learning network size arbitrary, only localized activations are of interest
Localized Excitations Network Structure spatial organization in a chain delta coupling equal synaptic weights ǫ only to the two nearest neighbours, all excitatory ǫ ǫ ǫ ǫ v i 2 v i 1 v i v i+1 v i+2 synaptic delay τ no synaptic changes, i.e. no learning network size arbitrary, only localized activations are of interest
Existence Theorem of Existence ( ) vreset v For τ > ln 1 + th v v th the spike-pattern below exists and is stable for [ 1 ( ǫ v th v + (v th v reset )e γ2τ) [,(1 e γτ )(v th v ) 2 unit 2 3 4 5 6 7 0 τ 2τ 3τ 4τ time
Existence Sketch of the Proof v i vth1 v 1 vreset1 vth2 v 2 vreset2 vth3 v 3 vreset3 vth4 v 4 vreset4 vth5 v 5 vreset5 vth6 v 6 vreset6 vth7 v 7 vreset7 vth8 v 8 vreset8 τ 2τ 3τ t
Existence Stability against Spike Time Perturbations If the crossing of v th is preserved, we have the Return map: ( ) ( ) ( t3,i+1 t3,i max(t3,i, t = F = 5,i ) + 2τ max(t 3,i, t 5,i ) + 2τ t 5,i+1 t 5,i ) No dependence of the concrete potentials because of the fix v reset
Existence Stability cont. membrane potentials while resynchronization v i vth3 v 3 vreset3 vth4 v 4 vreset4 vth5 v 5 vreset5 vth6 v 6 vreset6 τ τ + τ 2τ 2τ + τ 3τ + τ t
Simulation Onset via Input Weak input to a certain region initiates a persistent localized solution.
Simulation Spike Noise With a certain probability neurons fire spontaneously. t 10 9 50 8 7 6 5 30 4 3 2 10 1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 p spike *0.01 This can effect the width of the excitation.
Simulation Noisy Membrane Potential Additive white noise is applied to the membrane potentials. 15 15 10 10 5 5 0 0.5 1 1.5 2 2.5 3 0 1 2 3 4 5 6 7 2 1 0 1 1.5 1 0.5 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 0 1 2 3 4 5 6 7 Depending on the parameters this noise can have a positive or negative effect.
Simulation Membrane Noise 10 The border of the region of existence is softened. Weak noise kills localized excitations. Strong noise keeps them alive. 9 8 7 6 5 4 3 2 1 0 0 0.05 0.1 0.15 0.2
Feature Binding Model Feature Binding Model Actual problem in psychology: Feature Binding examplary setting in the visual pathway: Features of objects in the view are decomposed Shape Color This information has to be recombined Flexible Persistent for some time IT PFC V4 S. Becker, J. Lim (03) Journal of Cognitive Neuroscience
Feature Binding Model Feature Binding Model 3 chains of leaky integrate and fire neurons More complex intern neighborhood for strong localization global inhibtion Inter-layer connections are All to all equally excitatory between PFC V4 PFC IT IT PFC V4
Feature Binding Model Onset of a Flexible Link IT PFC V4 time vs. neuron index external input
Feature Binding Model IT 1 100 80 PFC 1 100 80 V4 1 100 80
Summary Conclusion Localized excitations occur even in very simple networks. They do not require: inhibition adaptation These localized excitations provide the flexibility to model feature binding in the visual system. Certain effects match the real situation: Ability for short term memory Limited memory capacity Wrong binding with a certain probability (hallucinations)
End Outlook Generalizing the notion of localization to consider networks of more complex structure. Inner neurons boundary neurons outer neurons Extending the Feature Binding model to two dimensions and apply biological relevant parameters to match psychological experiments.
End Acknowledgements Theo Geisel Michael Herrmann (Greetings from him!) Vincent David, Joachim Hass
Appendix Results of the feature-binding model Killing Memory IT 1 100 80 PFC 1 100 80 V4 1 100 80
Appendix Results of the feature-binding model Searching for a Lost Feature IT 1 100 80 PFC 1 100 80 V4 1 100 80
Appendix Results of the feature-binding model Changing a Feature IT 1 100 80 PFC 1 100 80 V4 1 100 80