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
Excitable Cells Outline Excitable cells Hybrid model for excitable cells Conclusions and future directions
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
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
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
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 1 3 0. 1 V V 20 + 1 ( 25. 01. V ) α m ( V ) = 25. 01. V e 1 β ( V ) = 4e m ( 01. 001. V ) α n ( V ) = 1 0. 1 V e 1 β ( V ) = 0125. 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
Excitable Cells Macro Analyzable but unrealistic Micro Realistic, but not analyzable. Simulation is slow.
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
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
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 25 20 APD 15 10 50 100 150 200 stimulation frequency
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
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 )
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]
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
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
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