The Effects of Hebbian Learning on the Structure and Dynamics of Chaotic Neural Networks

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The Effects of Hebbian Learning on the Structure
and Dynamics of Chaotic Neural Networks
H. Berry1, B. Cessac2, B. Delord 3, M. Quoy4, B.
Siri1
1 Alchemy, INRIA, Orsay, France 2 Institut Non
Linéaire de Nice Odyssee, INRIA,
Sophia-Antipolis , France 3 ANIM, Univ. PM
Curie, Paris , France 4 ETIS, Univ.
Cergy-Pontoise , France
2
Cortical circuits display complex structure
V2

• The local micro-connectivity structure of the
  cortex is massively recurrent
• At several length scales, the connectivity
  structure is complex and appears partly random

V1
LGN
3
Neurons show complex dynamics

• This complex recurrent structures yield a dense
  riche network activity even in the absence of
  stimulus ongoing (spontaneous) activity
• This ongoing activity
• depends on the recurrent complex structure
• is highly variable but not only noise Arieli et
  al (1995)
• conditions the evoked response to stimuli Arieli
  et al (1996)

Membrane potential of a cortical neuron in vivo
Lampl et al (1999)
4
Plasticity yields dynamic connectivity

• e.g. cortical rewiring in ferrets Sur Leamey
  (2001)

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Structure / Dynamics in Neural Networks

• Network topology conditions Neuron dynamics
• Synaptic plasticity (i.e. Hebbian learning)
  Neuron dynamics conditions Network topology
• ? mutual coupling between structure dynamics and
  neurons dynamics
• A complex dynamical system and key pbl in
  neuroscience
• But not really understood yet in the case of
  complex dynamics

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W01
0
2
W02
W03
3
W01
0
1
neuron state
connection weight
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Outline of the talk

• Aim Structure/dynamics relationships in random
  neural networks with rich dynamics and Hebbian
  plasticity
• A simple but generic model to allow rigorous
  analysis
• Structure changes at several length scales
• Effects on network dynamics
• Effects on network function
• Perspectives competition between Hebbian and
  intrinsic plasticity

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Chaotic random neural networks

• Firing rate dynamics cortex-like netwk structure
• Random sparse connectivity (15) Gabbott
  Somogyi (1986)
• 25 inhibitory neurons Markram et al (1997)

j
i

• Initial synaptic weights Wij(0) i.i.d. random
  var. set so that initial dynamics is chaotic
  (rich dynamics)

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Typical evolution of dynamics complexity
Learning an external input
N 500 neurons

• Hebbian learning generically decreases the
  richness of the dynamics

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Structure changes motifs

• Motifs patterns (sub-graphs) that recur more
  often than at random
• Motifs composition of a complex network its
  imprint (Milo et al, 2002)
• e.g. 3-node directed motifs
• In the cortex, some motifs are more frequent than
  by chance (Sporns Kötter, 2004)

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Motifs evolution by Hebbian plasticity
Statistical enrichment
Learning epochs

• Hebbian plasticity reorganizes the motifs
• Specific motifs are reinforced densely
  interconnected ones
• Dynamical consequences? (current works)
• No changes at early epochs, i.e. where most
  dynamical changes occur

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The small-world of the cortex

• Small-world more-than-by-chance triangle motifs
  (cliquishness) AND short mean shortest path
• Efficient information transfer both locally and
  globally
• Experimental quantification of topology at
  several length scales
• Neural networks grown in vitro physical and
  functional
• C. elegans neural network (physical)
• Cortico-cortical area physical connectivity in
  Cat, Macaque Human
• Human cortical functional connectivity via MEG,
  EEG or fMRI
• Human cortical layer-to-layer connectivity
• All display small-world structure (not scale-free
  except in Eguiluz et al. (2005) Phys Rev Lett,
  94018102)

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Hebbian Learning-induced SW
Clustering index
Mean shortest path
Learning epochs
Learning epochs

• Hebbian plasticity reorganizes the weights over
  the connectivity network as a small-world
  structure
• Again no changes at early epochs, i.e. where most
  dynamical changes occur

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Spectral properties

• Exponential decay of W norm

• Largest Lyapunov exponent

Hebbian learning yields reduction of dynamics
complexity through decay of W norm
L1
Learning epochs
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Pattern Sensitivity

• Quantitative measurement of how network dynamics
  changes when pattern is removed
• Hebbian learning leads the system close to a
  bifurcation point (edge of chaos) where pattern
  sensitivity is maximal

Learning epochs
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Summary

• Effects of Hebbian learning in recurrent neural
  networks
• drastic changes of the motifs/microcircuitry
• synaptic weights redistribute as a small world
  structure
• shrinks Jacobian weight matrix spectral radius
• this reduces the dynamics complexity entropy
  (KS)
• Functional effects
• leads the system close to the edge of chaos
  bifurcation
• where sensitivity to input pattern is maximal
• At long learning times constant firing frequency
  no response to pattern presence

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Perspectives intrinsic plasticity

• Activity dependent modification of the
  excitability (firing threshold, gain)
• No change of the synaptic weights
• Homeostatic effects

Firing frequency
Input current
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HebbianIntrinsic preliminary results

• Dynamics still rich at long times
• Sensitivity to the input pattern preserved

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Perspectives Regulations by glial cells

• Collaboration with E. Ben Jacob (Tel Aviv Univ.,
  Israel) and V. Volmann (UCSD Salk Institute, La
  Jolla, US).
• Understand the possible effects at the single
  neuron and network levels of neuronal
  excitability regulation by glial cells.

Field, Scientific American, 2004
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Thanks!!

• Further results / analysis in
• Siri et al. (2008) Neural Computation,
  202937-2966.
• Siri et al. (2007) Journal of Physiology
  (Paris),101136-148.
• http//www-rocq.inria.fr/hberry/

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The model RRNNs

• 2 different time scales Learning and dynamics
• Learning occurs every t dynamics steps
• Dynamics

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Generic Hebbian Learning

• Initial weights are random Wij(1) N(0, 1/N)
• A generic learning rule
• A Hebbian rule if
• h gt 0 for i (postsyn) j (presyn) both active
  during T (LTP)
• h lt 0 for i (postsyn) inactive j (presyn)
  active during T (LTD)
• h 0 whenever j (presyn) inactive during T
• For numerical simulations

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Bifurcations

• Consider the Jacobian matrix at x
• DFx spectrum can be contracted by
• Increased neuron saturation to 0 or 1
• The contraction of W spectrum
• Confirmed by numerical simulations

bifurcations
l 0.90
l 0.80
Hebbian learning leads the system through a
bifurcation