Lyapunov Functions and Memory Justin Chumbley. Why do we need more than linear analysis? What is Lyapunov theory? – Its components? – What does it bring?

Lyapunov Functions and Memory

Justin Chumbley

• Why do we need more than linear analysis?• What is Lyapunov theory? – Its components?– What does it bring?

• Application: episodic learning/memory

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

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

• Positive definite state function

e.g.

• Unique singular point at 0

• Not unique U

*

• U defines the valley– Do state trajectories travel downhill? Temporal

change of pd state function along trajectories?– Time implicit in U

e.g.

• N-dim case

Lyapunov functions and asymptotic stability

• Intuition– Water down a valley all trajectories in a

neighborhood approach singular point as

satisfies a

• 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

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

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.

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

• 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

Where does U come from?

• No general rule.• Another e.g. divisive feedback

*

Memory

• Declarative – Episodic– Semantic

• Procedural• …

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

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

• 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…

Theorem 14

For reference

Can be generalized to higher order

s

,

Pattern recognition (matlab)

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

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

Learning (matlab)

• One stimuli• Multiple stimuli

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