Learning Cycle Linear Hybrid Automata for Excitable Cells

Transcription

1 Learning Cycle Linear Hybrid Automata for Excitable Cells Sayan Mitra Joint work with Radu Grosu, Pei Ye, Emilia Entcheva, I V Ramakrishnan, and Scott Smolka HSCC 2007 Pisa, Italy

2 Excitable Cells Outline Excitable cells Hybrid model for excitable cells Conclusions and future directions

3 Excitable Cells Excitable Cells An excitable cell generates electrical pulses or action potentials in response to electrical stimulation Examples: neurons, cardiac cells, smooth muscle cells Local regeneration allows electric signal propagation without damping Building block for electrical signaling in brain, heart, and muscles Neurons of a squirrel University College London Artificial cardiac tissue University of Washington

4 Excitable Cells Interaction of Excitable Cells Action Potential (AP) depends on stimulus, membrane voltage of neighboring cells, state of cell itself Normal: synchronous pulses, spiral waves Abnormal: incoherent pulses, wave breakup Leads to cardiac arrhythmia, epilepsy voltage Stimulus Schematic Action Potential Threshold failed initiation Resting potential time

5 Macro Models of Action Potentials Cellular automata Oscillators and uniform coupling between cells [Kuramoto`84] Small world network of coupled oscillators [Watts & Strogatz`98] θi = ωi + t i = 1... N K N N j= 1 sin( θ θ ) j i

6 Excitable Cells Micro Models for Action Potentials Membrane potential for squid giant axon [Hodgkin Huxley`52] Luo Rudy model (1991) for cardiac cells of guinea pig Neo Natal Rat (NNR) model for cardiac cells of rat Na + K+. CV = g m h ( V V ) + g n ( V V ) + g ( V V ) + I. h 3 4 Na Na K K L L st m = ( α + β ) m + α. α ( V ) = 007. e β h ( V ) = e m m m h = ( α + β ) h + α. h h h n = ( α + β ) n + α n n n V V ( V ) α m ( V ) = V e 1 β ( V ) = 4e m ( V ) α n ( V ) = V e 1 β ( V ) = e n V 18 V 80 I Na I K I st Outside I L Na K L g Na g K g L C I C C V Large state space Nonlinear differential equations Multiple spatial and temporal scales V Na V K V L Inside

7 Excitable Cells Macro Analyzable but unrealistic Micro Realistic, but not analyzable. Simulation is slow.

8 Hybrid Automata Model for Excitable Cells Linear Hybrid Approximations for Action Potentials Suppose, AP can be partitioned into modes so that in mode M, v can be approximated by: x i = b Mi x i v = S i x i, M e {S,U,E,P,F,R} b Mi s can be found by Prony s method which fits sum of exponentials to data Mode switches at the beginning and end of stimulus when v crosses threshold voltages V M But, stimulus can appear at any M State of cell at the time of arrival of stimulus influences behavior of cell for the next AP b Mi s history dependent voltage (v) Upstroke (U) Stimulated (S) Early Repol (E) v V U U Plateau (P) E time v V E Final Repol (F) P v V F v V R Resting (R) F S R

9 Hybrid Automata Model for Excitable Cells History Dependence of APs Frequency of stimulation determines voltage (v 0 )at the time of appearance of stimulus, which influences shape of next AP voltage Lower frequency: longer resting time and v 0 closer to resting voltage results in longer AP time v 0 Higher frequency: shorter AP

10 Hybrid Automata Model for Excitable Cells History Dependence of APs Frequency of stimulation determines voltage (v 0 )at the time of appearance of stimulus, which influences shape of next AP voltage v 0 Even higher frequency: conjoined AP, bifurcation time APD stimulation frequency

11 Hybrid Automata Model for Excitable Cells History Dependence and Restitution Frequency of stimulation determines voltage (v 0 )at the time of appearance of stimulus, which influences shape of next AP voltage AP Duration (APD) Diastolic Interval (DI) Restitution curve: APD vs. DI Slope > 1 indicates breakup of spiral waves under high frequency stimulation Local to global behavior 10% of peak time v 0

12 Hybrid Automata Model for Excitable Cells Cycle Linear Hybrid Automata (CLHA) Uncountable family of modes = μ : Epoch and Regime (, <) is a total order Linear dynamics in each mode Unique ge that is visited infinitely many times M x i = b Mi (v 0 ) x i v = S i x i M e {S,U,E,P,F,R} There exists a snapshot function : Xö, such that for any switch (x 1, e 1, r 1 ) ö (x 2, e 2, r 2 ) (i) r 2 = g and e 2 = (x 1 ), or (ii) r 2 g, e 2 = e 1 and r 2 <r 1 = {S,U,E,P,F,R} determined by v 0 g v V T (v 0 ) U v 0 := v E S v V E (v 0 ) v 0 := v P v 0 := v v 0 := v R v V F (v 0 ) F v V R (v 0 )

13 Hybrid Automata Model for Excitable Cells Identifying CLHA Parameters for a single AP For each mode, we seek a solution for LTI: x& = bx, x(0) = a, b= diag( b,..., b ), a = [ a... a ] v = n i= 1 1 n 1 x i n T Observable solution is a sum of v = n i= 1 a e i bt i exponentials: Curve segments are Convex, concave or both Consequences: Solutions: might require at least two exponentials Coefficients a i and b i : positive/negative or real/complex Exponential fitting: Modified Prony s method [Osborne and Smyth `95]

14 Hybrid Automata Model for Excitable Cells Parameters as Functions of History Variable V TVE Parameters: Threshold voltages V S, V E, Coefficients of differential equations b S1, b S2, b E1, Coefficients in reset maps From each stimulation frequency in the training set, we get a corresponding value for b S1, b S2,, V S, V E, Apply Prony s method (a second time) to obtain b S1 as a function of v 0 : b M1 (v 0 ) = c M1 exp (v 0 d M1 ) + c M1 exp (v 0 d M1 ), for each M e {S,U,E,P,F,R} V T (v 0 ) = c T exp (v 0 d T ) + c T exp (v 0 d T ) v 0 V P V L

15 Conclusions: Results Contributions and Simulation Results CLHA as a model for almost periodic systems Iterative process to obtain excitable cell model with desired accuracy Simulation efficiency (> 8 times faster) [True, Entcheva, et al.] Biological interpretation of state variables x 1, x 2 ; restitution curve Spiral wave generation and breakup Spiral waves Breakup

16 Conclusions: Future Directions Future Directions CLHA for stimulation with different shapes CLHAs coupled through pulses or diffusion for analyzing synchronization conditions Specification of spatiotemporal voltage patterns Distributed control through targeted stimulation