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Lyapunov Functions and Memory
Justin Chumbley
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• Why do we need more than linear analysis?• What is Lyapunov theory? – Its components?– What does it bring?
• Application: episodic learning/memory
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Linearized stability of non-linear systems: Failures
• Is there a steady state under pure imaginary eigenvalues?– theorem 8 doesn’t say
• Size/Nature of Basin of attractions?– cf a small neighborhood of the ss (linearizing)
• Lyapunov– Geometric interpretation of state-space
trajectories
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Important geometric concepts(in 2d for convenience)
• State function– scalar function U of system with continuous partial
derivatives
– A landscape• Define a landscape with steady state at the bottom of a
valley
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• Positive definite state function
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e.g.
• Unique singular point at 0
• Not unique U
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*
• U defines the valley– Do state trajectories travel downhill? Temporal
change of pd state function along trajectories?– Time implicit in U
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e.g.
• N-dim case
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Lyapunov functions and asymptotic stability
• Intuition– Water down a valley all trajectories in a
neighborhood approach singular point as
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satisfies a
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• Ch 8 Hopf bifurcation• Van der Pol model for a heart-beat
– Analyzed at bifurcation point (where linearized eigenvalues are purely imaginary)
– At this point…
(0,0) is the only steady state Linearized analysis can’t be applied (pure imaginary eigs)– But: pd state function has time derivates along trajectories
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satisfies b
• So– Except on x,y axes where – But when x = 0 then
trajectories will move to pointswhere
- So U is a Lyapunov function for …
- Ss at (0,0) is asymptotically stable
Conclusion: have proven stability where linearization fails
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Another failure of Theorem 8
• Points ‘sufficiently close’ to asymptotically stable steady state go there as
• But U defines ALL points in the valley in which the ss lies! – Intuition: any trajectory starting within the valley
flows to ss.
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Formally
• many steady and basins– Assume we have U for
• It delimits a region R within which theorem 12 holdsA constraint U<K defines a subregion within the basin
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• Key concept: closed contour (or spheroid surface in 3d+) that encloses the ss– As long as this region is within R, T12 guarantees
that all points go to steady state– K = highest point on valley walls from which
nothing can flow out– is a lower bound on the basin ( depends on U
too!) e.g. use
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Where does U come from?
• No general rule.• Another e.g. divisive feedback
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*
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Memory
• Declarative – Episodic– Semantic
• Procedural• …
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Episodic memory (then learning)• Computational level: one-shot pattern learning & robust recognition
(Generalization over inputs and discriminate) – Learn to generalize/discriminate appropriately, given our uncertainty (statistics)– p(f,x) ? p((x) ? …. e.g. regresion/discriminant
• Algorithmic level: use stable dynamic equilibria – (x) is steady-state of system m, given initial condition x
– not smooth generalization (over inputs)– Dynamics
• Implementation level constraints• Anatomical: Hippocampal ca3 Network • Physiological: Hebbian
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m
• 16*16 pyramidal – Completely connected but not self-connected
• 1 for feedback inhibition
If R is a rate/speed, then acceleration of R is a sigmoidal function of PSPNo Self connection … is pre-learntPSP includes inputs: a subset x of neurons exogenously stimulatedWhat is (x) go?Sigma = semi-saturation time constant
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• Aim– Understand generalization/discrimination
• Strategy– Input in the basin will be ‘recognized’ • i.e. identified with the stored pattern (asympotically)
– Lyapunov theory assess basins of attraction
Notation:etc…
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Theorem 14
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For reference
Can be generalized to higher order
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s
,
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Pattern recognition (matlab)
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Hebb Rule
• Empirical results– Implicate cortical and hippocampal NMDA – 100-200ms window for co-occurance– Presynaptic Glu and Postsynaptic depolarisation
by backpropogation from postsynaptic axon (Mg ion removal).
Chemical events change synapse
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For simplicity…
• M = max firing rate – (both pre and post must be firing higher than half maximum)
• Synapse changes to fixed k when modified• Irreversible synaptic change• All pairs symmetrically coupled
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Learning (matlab)
• One stimuli• Multiple stimuli
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Pros and limitations of Lyapunov theory
• More general stability analysis • Basins of attraction• Elegance and power
• No algorithm for getting U• Not unique U: each gives lower bound on
basin
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