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Functional resilience in neural networks
Mediterranean School of Complex Networks
Edward Laurence September 5, 2017
Département de physique, de génie physique, et d’optique Université Laval, Québec, Canada
Functional resilience in neural networks Mediterranean School of - - PowerPoint PPT Presentation
Functional resilience in neural networks Mediterranean School of - - PowerPoint PPT Presentation
Functional resilience in neural networks Mediterranean School of Complex Networks Edward Laurence September 5, 2017 Dpartement de physique, de gnie physique, et doptique Universit Laval, Qubec, Canada 0 Resilience Ability to
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2
The brain is resilient Plasticity, compensation, ... The details and strategies are still unknown
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3 Connectomics
Nodes Neurons of activity xi(t) Edges Synapses of weight wij(t) Perturbations ∆xi(t), ∆wij(t), edges/nodes removal, ...
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4 TOC
Effective formalism Description Application to neural networks Approximations and errors Adaptive connectivity Special behaviors Measures of resilience
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5 Effective formalism | Description
Effective formalism Presented by Gao et al. 2016
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6 Effective formalism | Description
N-dimensional complete system
- xi F(xi) +
N
- j1
wijG(xi, xj)
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6 Effective formalism | Description
N-dimensional complete system
- xi F(xi) +
N
- j1
wijG(xi, xj)
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6 Effective formalism | Description
N-dimensional complete system
- xi F(xi) +
N
- j1
wijG(xi, xj) 1-dimensional effective system
- xeff F(xeff) + βeffG(xeff, xeff)
xeff L x
- ij wijxj
- ij wij
; βeff L s
- ijk wijwjk
- ij wij
L x Neighborhood average of x
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7 Effective formalism | Neural networks
1-dimensional effective system
- xeff F(xeff) + βeffG(xeff, xeff)
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7 Effective formalism | Neural networks
1-dimensional effective system
- xeff F(xeff) + βeffG(xeff, xeff)
Neural dynamics - Hopfield model
- xi −xi +
- j
wijσ
- λ
xj − µ σ(y) 1 1 + e−y
3 6 9 y 0.0 0.2 0.4 0.6 0.8 1.0 σ(y)
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7 Effective formalism | Neural networks
1-dimensional effective system
- xeff F(xeff) + βeffG(xeff, xeff)
Neural dynamics - Hopfield model
- xi −xi +
- j
wijσ
- λ
xj − µ σ(y) 1 1 + e−y
3 6 9 y 0.0 0.2 0.4 0.6 0.8 1.0 σ(y)
- xeff −xeff + βeffσ
- λ
xeff − µ
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8 Effective formalism | Neural networks
- xeff −xeff + βeffσ
- λ
xeff − µ Stationnary state xeff 0
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8 Effective formalism | Neural networks
- xeff −xeff + βeffσ
- λ
xeff − µ Stationnary state xeff 0
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8 Effective formalism | Neural networks
- xeff −xeff + βeffσ
- λ
xeff − µ Stationnary state xeff 0
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8 Effective formalism | Neural networks
- xeff −xeff + βeffσ
- λ
xeff − µ Stationnary state xeff 0
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8 Effective formalism | Neural networks
- xeff −xeff + βeffσ
- λ
xeff − µ Stationnary state xeff 0
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8 Effective formalism | Neural networks
- xeff −xeff + βeffσ
- λ
xeff − µ Stationnary state xeff 0
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9 Effective formalism | Neural networks
- xeff −xeff + βeffσ
- λ
xeff − µ Stationnary state xeff 0
(a)
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10 Effective formalism | Neural networks
- xeff −xeff + βeffσ
- λ
xeff − µ Good approximation for Homogeneous network Low inhibition High reciprocity wij wji
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Adaptive connectivity
11
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12 Plasticity
Resilience Ability to recover the original state in a reasonnable short period of time.
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12 Plasticity
Resilience Ability to recover the original state in a reasonnable short period of time. Modified Hebb’s rule
- wij τ−1
S (σiσj − wijσ2 j )
; σi σ[λ(xi − µ)]
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13 Recuperation
- wij τ−1
S (σiσj − wijσ2 j )
; σi σ[λ(xi − µ)] 2 3 4 5 6 7 8 9 10
βeff
2 4 6 8 10
xeff
τ −1
S
= 1 τ −1
S
= 0.7 τ −1
S
= 0.68 τ −1
S
= 0.3
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14 Measures of resilience
How to quantify resilience?
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14 Measures of resilience
How to quantify resilience? Recovery time Energy of recuperation Maximum damage Sensibility dxeff dβeff
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15 Measures of resilience | Time of recuperation
Recovery time Time to return in the surroundings of the original state
No recovery
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16 Measures of resilience | Time of recuperation
Recovery time Time to return in the surroundings of the original state
No recovery
Critical slowing down
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Conclusion
17
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18 Conclusion
Effective formalism Simple to use on neural dynamics. Valid for homogeneous, low inhibition and high reciprocity.
2 3 4 5 6 7 8 9 10
βeff
2 4 6 8 10
xeff
τ −1
S
= 1 τ −1
S
= 0.7 τ −1
S
= 0.68 τ −1
S
= 0.3
Resilience Introduce adaptive connectivity Recovery time is a good indicator of catastrophe
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