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René Doursat
Department of Computer Science & EngineeringUniversity of Nevada, Reno
Spring 2006
CS 790R SeminarModeling & Simulation
Neural Networks 1 –Synchronization in
Spiking Neural Networks
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Synchronization in Spiking Neural Networks
1. Temporal Coding2. Coupled Oscillators3. Synfire Chains
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Synchronization in Spiking Neural Networks
1. Temporal Coding• Neural networks
• The neural code
• Questions of representation
2. Coupled Oscillators3. Synfire Chains
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Synchronization in Spiking Neural Networks
1. Temporal Coding• Neural networks
– Structure of neural networks– Structure of a neuron– Propagation of a “spike”– Model of neural network
• The neural code
• Questions of representation
2. Coupled Oscillators3. Synfire Chains
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Neural networksStructure of neural networks
Phenomenonneurons together form... the brain! (and peripheral nervous system)
perception, cognition, actionemotions, consciousnessbehavior, learningautonomic regulation: organs, glands
Medial surface of the brain (Virtual Hospital, University of Iowa)
~1011 neurons in humanscommunicate with each other through (mostly) electrical potentialsneural activity exhibits specific patterns of spatial and temporal synchronization (“temporal code”)
Cortical layers
Pyramidal neurons andinterneurons
(Ramón y Cajal 1900)
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A typical neuron (http://www.bio.brandeis.edu/biomath/mike/AP.html)
Pyramidal neurons andinterneurons (Ramón y Cajal 1900)
Ionic channels opening and closing→ depolarization of the membrane
(http://www.awa.com/norton/figures/fig0209.gif)
Neural networksStructure of a neuron
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Propagation of the depolarization along the axon→ called “action potential”, or “spike”(http://hypatia.ss.uci.edu/psych9a/lectures/lec4fig/n-action-potential.gif)
(http://www.bio.brandeis.edu/biomath/mike/AP.html)
Neural networksPropagation of a “spike”
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Schematic neurons (adapted from CS 791S “Neural Networks”, Dr. George Bebis, UNR)
A binary neural network
Mechanismeach neuron receives signals from many other neurons through its dendritesthe signals converge to the soma (cell body) and are integratedif the integration exceeds a threshold, the neuron fires a spike on its axon
Neural networksModel of neural network
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1. Temporal Coding• Neural networks
• The neural code– Rate vs. temporal coding– Synchronization and correlations– Interest for temporal coding
• Questions of representation
2. Coupled Oscillators3. Synfire Chains
Synchronization in Spiking Neural Networks
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• Rate coding: average firing rate (mean activity)
• Temporal coding: correlations, possibly delayed
von der Malsburg, C. (1981) The correlation theory of brain function. Internal Report 81-2, Max Planck Institute for Biophysical Chemistry, Göttingen.
The neural codeRate vs. temporal coding
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low activity rate
low activity rate
low activity rate
high activity rate
high activity rate
high activity rate
1 and 2 more in sync than 1 and 3
4, 5 and 6 correlated through delays
The neural codeSynchronization and correlations
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• Historical motivation for rate coding– Adrian (1926): the firing rate of mechanoreceptor neurons in frog leg
is proportional to the stretch applied– Hubel & Wiesel (1959): selective response of visual cells; e.g., the
firing rate is a function of edge orientation
→ rate coding is confirmed in sensory system and primary cortical areas,however increasingly considered insufficient for integrating the information
• Recent temporal coding “boom”: a few milestones– von der Malsburg (1981): theoretical proposal to consider correlations– Abeles (1982, 1991): precise, reproducible spatiotemporal spike
rhythms, named “synfire chains”– Gray & Singer (1989): stimulus-dependent synchronization of
oscillations in monkey visual cortex– O’Keefe & Recce (1993): phase coding in rat hippocampus supporting
spatial location information– Bialek & Rieke (1996, 1997): in H1 neuron of fly, spike timing conveys
information about time-dependent input
The neural codeInterest for temporal coding
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1. Temporal Coding• Neural networks
• The neural code
• Questions of representation– The “binding problem”– Feature binding in cell assemblies– “Grandmother” cells– Relational graph format– Solving the binding problem with temporal coding– A molecular metaphor
2. Coupled Oscillators3. Synfire Chains
Synchronization in Spiking Neural Networks
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=
=
=
complex feature cells
=input
Questions of representationThe “binding problem”
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+ =
→ unstructured lists of features lead to the “superposition
catastrophe”
big
green
corners
soft
red
hard
small
Questions of representationFeature binding in cell assemblies
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...... ...... ......
+ =
→ one way to solve the confusion: introduce overarching complex
detector cells
Questions of representation“Grandmother” cells
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......... ...
. . . however, this soon leads to an unacceptable
combinatorial explosion!
Questions of representation“Grandmother” cells
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+ =
→ another way to solve the confusion:
represent relational information
Questions of representationRelational graph format
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=
=
=
complex feature cells
=input
→ another way to solve the confusion:
represent relational information
von der Malsburg, C. (1981) The correlation theory of brain function.
Questions of representationSolving the binding problem with temporal coding
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C3 H8 O 1-propanol
2-propanol
“cognitiveisomers” made of the same
atomic features
Questions of representationA molecular metaphor
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1. Temporal Coding• Neural networks
• The neural code
• Questions of representation
2. Coupled Oscillators3. Synfire Chains
Synchronization in Spiking Neural Networks
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1. Temporal Coding2. Coupled Oscillators3. Synfire Chains
Synchronization in Spiking Neural Networks
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1. Temporal Coding2. Coupled Oscillators
• Temporal tagging
• Group synchronization
• Traveling waves
3. Synfire Chains
Synchronization in Spiking Neural Networks
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1. Temporal Coding2. Coupled Oscillators
• Temporal tagging– The binding problem in language– A model of semantic binding: SHRUTI– Using correlations to implement binding
• Group synchronization
• Traveling waves
3. Synfire Chains
Synchronization in Spiking Neural Networks
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Temporal taggingThe binding problem in language
(b) Mary gives a book to John.(c)* Book John Mary give.
book
Mary
give
John
car
lamp
Rextalk
see
book
Mary
give
John
(a) John gives a book to Mary.
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Temporal taggingA model of semantic binding: SHRUTI
Shastri, L. & Ajjanagadde, V. (1993) From simple associations to systematic reasoning.Behavioral and Brain Sciences, 16(3): 417-451.
“John gives a book to Mary.”
... therefore: “Mary can sell the book.”
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Temporal taggingUsing correlations to implement binding
Binding by correlations, or “phase-locking”
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Temporal taggingUsing correlations to implement binding
Inference by propagation of bindings
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1. Temporal Coding2. Coupled Oscillators
• Temporal tagging
• Group synchronization– The scene segmentation problem– Excitatory-inhibitory relaxation oscillator– Van der Pol relaxation oscillator– Networks of coupled oscillators– A model of segmentation by sync: LEGION
• Traveling waves
3. Synfire Chains
Synchronization in Spiking Neural Networks
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scene analysis and segmentation is a fundamental aspect of perceptionability to group elements of a perceived scene or sensory field into coherent clusters or objectscan be addressed with temporal correlations, especially:dynamics of large networks of coupled neural oscillatorshow does it work? . . .
Group synchronizationThe scene segmentation problem
Schematic sceneWang, DeLiang (http://www.cse.ohio-state.edu/~dwang/)
Real sceneDoursat, Rene (http://www.cse.unr.edu/~doursat)
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Nexcitatoryneurons
Minhibitoryneurons
wEE
wIE
wEI
Group synchronizationExcitatory-inhibitory relaxation oscillator
relaxation oscillators exhibit discontinuous jumpsdifferent from sinusoidal or harmonic oscillations
Wang, DeLiang (http://www.cse.ohio-state.edu/~dwang/)
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Van der Pol relaxation oscillatorWang, DeLiang (http://www.cse.ohio-state.edu/~dwang/)
Group synchronizationVan der Pol relaxation oscillator
⇔
limit cycle attractor
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(a)01
-1.7
2
(b)
( ) ( )( )⎪⎩
⎪⎨⎧
+−−=
+−++++−= ∑η
η
cbvuav
Iuukzvuucu
iii
ij ijiiii
&
& 33
Oscillators and excitable unitsBonhoeffer-Van der Pol (BVP) stochastic oscillator
two activity regimes: (a) sparse stochastic and (b) quasi periodic
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Group synchronizationNetworks of coupled oscillators
Wang, DeLiang (http://www.cse.ohio-state.edu/~dwang/)
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locally coupledglobally coupledindirectly coupledthrough central pacemaker
Terman & D.L. Wang’s (1995)LEGION network: Locally ExcitatoryGlobally Inhibitory Oscillator Network(http://www.cse.ohio-state.edu/~dwang/)
global inhibitor
Group synchronizationA model of segmentation by sync: LEGION
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Group synchronizationA model of segmentation by sync: LEGION
Wang, D. L. & Terman, D. (1995) Locally excitatory globally inhibitory oscillator networks. IEEE Trans. Neural Net., 6: 283-286.
achieving fast synchronization with local, topological coupling only
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Group synchronizationA model of segmentation by sync: LEGION
Wang, D. L. & Terman, D. (1997) Image segmentation based on oscillatory correlation. Neural Computation, 9: 805-836,1997
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Group synchronizationA model of segmentation by sync: LEGION
Wang, D. L. & Terman, D. (1997) Image segmentation based on oscillatory correlation. Neural Computation, 9: 805-836,1997
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1. Temporal Coding2. Coupled Oscillators
• Temporal tagging
• Group synchronization
• Traveling waves– Phase gradients, instead of plateaus– Wave propagation and collision
3. Synfire Chains
Synchronization in Spiking Neural Networks
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Traveling wavesPhase gradients, instead of plateaus
x
ϕπ
-π
x
ϕπ
-π
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“Grass-fire” wave on 16x16 network of coupled Bonhoeffer-van der Pol units
Traveling wavesDetail
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Traveling wavesWave propagation and collision
Doursat, R. & Petitot, J. (2005) Dynamical Systems and Cognitive Linguistics: Toward an Active Morphodynamical Semantics. IJCNN’05, to appear in Neural Networks.
64 x 64 lattice of locally coupled Bonhoeffer-van der Pol oscillators
t = 32t = 18t = 5
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(a) (b)
t = 22 t = 34t = 5
Two cross-coupled, mutually inhibiting lattices of coupled oscillators
Traveling wavesWave propagation and collision
Doursat, R. & Petitot, J. (2005) Dynamical Systems and Cognitive Linguistics: Toward an Active Morphodynamical Semantics. IJCNN’05, to appear in Neural Networks.
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1. Temporal Coding2. Coupled Oscillators
• Temporal tagging
• Group synchronization
• Traveling waves
3. Synfire Chains
Synchronization in Spiking Neural Networks
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1. Temporal Coding2. Coupled Oscillators3. Synfire Chains
Synchronization in Spiking Neural Networks