Hybrid Automata and ɛ-analysis on a Neural Oscillator

Transcription

1 Hybrid Automata and ɛ-analysis on a Neural Oscillator A. Casagrande 1 T. Dreossi 2 C. Piazza 2 1 DMG, University of Trieste, Italy 2 DIMI, University of Udine, Italy

2 Intuitively... Motivations: Reachability Analysis of Hybrid Automata with applications to Biological Models Automatic analysis (... as much as possible) Exploit existing approximation techniques Avoid ad-hoc studies Case of study: Neural oscillator Rhythmic neural activity Interaction between neurons

3 Intuitively... Saddle point Avoid ad-hoc studies

4 Outline 1 Hybrid Automata 2 Approximation Techniques 3 Neural Oscillator Model 4 Computations 5 Conclusions

5 Hybrid Automata Syntax Finite state automaton H + Continuous variables Z, Z State: pair v, r Z 1 21 On Z 1 =Z 1 Off Z 1 22 Z 1 =Z 1 e T Z 1 18 Z 1 =Z 1 e T Z 1 19 Z 1 =Z 1 Example: Thermostat

6 Hybrid Automata Semantics - Continuous Transition: v, r t C v, s Continuous trajectory: f v : R d(h) (R + R d(h) ) where Dyn(v)[Z, Z, T ] is Z = f v (Z)(T ) r = f v (r)(0), s = f v (r)(t) Invariant: Inv(v)[f v (r)(t )] for each t [0, t] Z 1 21 On Z 1 =Z 1 Off Z 1 22 Z 1 =Z 1 e T Z 1 18 Z 1 =Z 1 e T Z 1 19 Z 1 =Z 1

7 Hybrid Automata Semantics - Discrete Transition: v, r e D v, s Discrete jump: e = v, v Activation: Act(e)[r] Reset: Res(e)[r, s] Z 1 21 On Z 1 =Z 1 Off Z 1 22 Z 1 =Z 1 e T Z 1 18 Z 1 =Z 1 e T Z 1 19 Z 1 =Z 1

8 Techniques Reachability Hybrid Automata Reachability Undecidable! [HKPV95] Approximation Techniques: Noise and Disturbed Automata [Frä99] Approximate Bisimulations and Simulations [GP07] ɛ-semantics [CPP09]... Template Polyhedral [SDI08] Abstractions for Hybrid Systems [Tiw08]

9 Techniques Noise and Disturbed Automata [Frä99] Continuous variables provide unbounded quantity of memory Trajectories of real systems are subject to noise Definition H is a disturbance variant of H of noise level ɛ or more if for each pair of states s, s such that δ(s, s t ) < ɛ it holds that if r C s in t H, then r C s in H. A hybrid automaton H is robust if there exists an ɛ s.t. if a does not reach a in H, then a does not reach a in H Noise ensures (semi-)decidability of reachability problem

10 Techniques Approximate Bisimulations and Simulations [GP07] Relaxation of bisimulations and simulations Simplification of dynamics and resets Observation map: : R d(h) R d Z T Trajectory 1 Trajectory 2 ɛ-bisimulation

11 Techniques ɛ-semantics [CPP09] Infinite precision is an approximation of reality Semantics of a formula ψ is the set of points { ψ } R n which satisfies ψ Formulæ semantics of dimension of at least ɛ B(S, ɛ) = {q R n p S(δ(p, q) < ɛ)} Example (Sphere Semantics) def ( t 1 t 2 ) ɛ = B({ t 1 t 2 }, ɛ), for {=, <} def ( ψ 1 ψ 2 ) ɛ = B({p}, ɛ) B({p},ɛ) ( ψ 1 ) ɛ ( ψ 2 ) ɛ def ( X ψ[x, Z] ) ɛ = ( ψ[r, Z] ) ɛ r R

12 Techniques Comparison Noise ɛ-bisimulation ɛ-semantics

13 Techniques Computing ɛ-semantics Translation for any ϕ[x ] T and ɛ R >0 (ϕ) ɛ [X ] T such that ( ϕ[x ] ) ɛ = { (ϕ) ɛ [X ] } Expansion Example ((t 1 def t 2)[Y, W ]) ɛ = W 0((t 1 t 2)[Y, W 0] δ(w 0, W ) < ɛ), for {=, <} (φ[y, W def ] ψ[y, W ]) ɛ = W 0( W 1(δ(W 0, W 1) < ɛ ( (φ) ɛ (ψ) ɛ )[Y, W 1]) δ(w 0, W ) < ɛ) ( X φ[y def, X, W ]) ɛ = W 0( W 1(δ(W 0, W 1) < ɛ X 0( (φ[y, X0, W 1]) ɛ ) δ(w 0, W ) < ɛ) Granularity

14 Neural Oscillator Model Xe = Xe τ + tanh(λ Xe) tanh(λ Xe) Continuous model [TMBD99] f (τ, λ) : Xi = Xi + tanh(λ Xi) + tanh(λ Xi) τ Piecewise model fα(τ, λ) : Xe = Xe τ + h λ,α(xe) h λ,α (Xi) Xi = Xi τ + h λ,α(xe) + h λ,α (Xi) h λ,α (X ) approximating tanh(λ X )

15 Neural Oscillator Hybrid Automaton Location Activation Hybrid Automaton associated to f H f α(τ, λ)

16 Neural Oscillator Experimental results ɛ-noise: 0, 0 is an unstable equilibrium H f is fragile 0, 0 in H f is different from 0, 0 in H f It is not possible to define robust models Unsafe ɛ-(bi)simulations: ɛ-simulation Reduction of analysis complexity 0, 0 is a singularity of the model: 0, 0 neighborhood reaches the limit cycle No ɛ-(bi)simulation relates 0, 0 with any other state

17 ɛ-semantics: Neural Oscillator Experimental results 1 Approximating (polynomials) solutions of differential equations 2 Translating ɛ-semantics (Sphere Semantics) 3 Computing exploiting cylindrical algebraic decomposition 6 Trajectory from (-0.5,-0.5) Trajectory from (-1.0,6.0) Trajectory from (0,0) Trajectory from (-1,6) Original evolution First degree approximation evolution

18 Neural Oscillator Experimental results ɛ-semantics (formally and automatically proves that): Flow tube including limit cycle 0, 0 reaches the limit flow tube No need to modify the automaton (syntax) but... ɛ-semantics translation increases fromulæ complexity Direct computation of reach set high complexity Reformulation of the problem in form of a property: Convergence to the limit cycle

19 Convergence and Formulæ reduction Q 1 X e > ɛ P 1 X 0 P 0 Q 0 X i < ɛ Π 1 Convergence Formula reduction

20 Conclusions Conclusions: ɛ-semantics better reflects the real system behaviour... high computational complexity Future works: Combine several hybrid automata Integrations of numerical approximation techniques Automatic simplification of the formulæ

21 References I A. Casagrande, C. Piazza, and A. Policriti. Discrete semantics for hybrid automata. Discrete Event Dyn. Syst., 19(4): , M. Fränzle. Analysis of hybrid systems: An ounce of realism can save an infinity of states. In Proc. of Workshop on Computer, Science, and Logic (CSL 99), volume 1683 of LNCS, pages Springer, A. Girard and G. J. Pappas. Approximation metrics for discrete and continuous systems. IEEE Trans. Automat. Control, 52(5): , May 2007.

22 References II T. A. Henzinger, P. W. Kopke, A. Puri, and P. Varaiya. What s decidable about hybrid automata? In Proc. of ACM Symposium on Theory of Computing (STOCS 95), pages , S. Sankaranarayanan, T. Dang, and F. Ivancic. Symbolic model checking of hybrid systems using template polyhedra. In Proc. of Conference on Tools and Algorithms for the Construction and Analysis of Systems (TACAS 08), volume 4963 of LNCS, pages Springer, A. Tiwari. Abstractions for hybrid systems. Formal Methods in System Design, 32(1):57 83, 2008.

23 References III A. Tonnelier, S. Meignen, H. Bosch, and J. Demongeot. Synchronization and desynchronization of neural oscillators. Neural Networks, 12(9): , 1999.