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Learning Cycle-Linear Hybrid Automata for Excitable Cells

Explore the use of cycle-linear hybrid automata to model excitable cells like neurons, cardiac, and muscular cells, revealing their complex dynamics and aiding in formal analysis and control. The project aims to build linear hybrid automata models of excitable cell behavior, synthesize controllers, and validate them in vitro to potentially impact cardiovascular and neurological diseases. Mathematical models like the Hodgkin-Huxley and Luo-Rudy models provide frameworks for understanding excitable cell behavior and action potentials, crucial for advancing research in this field.

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Learning Cycle-Linear Hybrid Automata for Excitable Cells

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  1. Learning Cycle-Linear Hybrid Automata for Excitable CellsRadu GrosuSUNY at Stony Brook Joint work with Sayan Mitra and Pei Ye

  2. Motivation • Hybrid automata: an increasingly popular formalism for approximating systems with nonlinear dynamics • modes: encodevarious regimes of the continuous dynamics • transitions: express the switching logic between the regimes • Excitable cells: neuronal, cardiac and muscular cells • Biologic transistors whose nonlinear dynamics is used to • Amplify/propagate an electrical signal (action potential AP)

  3. Motivation • Excitable cells (EC) are intrinsically hybrid in nature: • Transmembrane ion fluxesand AP vary continuously, yet • Transition from resting to excited states is all-or-nothing • ECs modeled with nonlinear differential equations: • Invaluable asset to reveal local interactions • Very complex: tens of state vars and hundreds of parameters • Hardly amenable to formal analysis and control

  4. Project Goals • Learn linear HA modeling EC behavior (AP): • Measurements readily available in large amounts • Analyze HA to reveal properties of ECs: • Setting up new experiments for ECs may take months • Synthesize controllers for ECs from HA: • Higher abstraction of HA simplifies the task • Validate in-vitro the EC controllers: • Cells grown on chips provided with sensors and actuators

  5. Impact • 1 million deaths annually: • caused by cardiovascular disease in US alone, or • more than 40% of all deaths. • 25% of these are victims of ventricular fibrillation: • many small/out-of-phase contractions caused by spiral waves • Epilepsy is a brain disease with similar cause: • Induction and breakup of electricalspiral waves.

  6. Ventricular Tachycardia / Fibrillation

  7. Mathematical Models • Hodgkin-Huxley (HH) model (Nobel price): • Membrane potential forsquid giant axon • Developed in 1952. Framework for the following models • Luo-Rudy (LRd) model: • Model forcardiac cells of guinea pig • Developed in 1991. Much more complicated. • Neo-Natal Rat (NNR) model: • Being developed at Stony Brook by Emilia Entcheva • In-vitro validation framework. Very complicated, too.

  8. K+ Na+ Outside C Na K L Inside Active Membrane Conductances vary w.r.t. time and membrane potential

  9. Action Potential

  10. Ist Outside INa IK IL IC gNa gK gL C V VNa VK VL Inside Currents in an Active Membrane

  11. Ist Outside INa IK IL IC gNa gK gL C V VNa VK VL Inside Currents in an Active Membrane

  12. Ist Outside INa IK IL IC gNa gK gL C V VNa VK VL Inside Currents in an Active Membrane

  13. Ist Outside INa IK IL IC gNa gK gL C V VNa VK VL Inside Currents in an Active Membrane

  14. Ist Outside INa IK IL IC gNa gK gL C V VNa VK VL Inside Currents in an Active Membrane

  15. Kinetics of a Gate Subunit

  16. Kinetics of a Gate Subunit

  17. Kinetics of a Gate Subunit

  18. The Full Hodgkin-Huxley Model

  19. vn Frequency Response APD90: AP > 10% APmDI90: AP < 10% APmBCL:APD + DI

  20. vn Frequency Response APD90: AP > 10% APmDI90: AP < 10% APmBCL:APD + DI S1S2 Protocol: (i) obtainstable S1;(ii) deliverS2 with shorter DI

  21. Frequency Response APD90: AP > 10% APmDI90: AP < 10% APmBCL:APD + DI S1S2 Protocol: (i) obtainstable S1;(ii) deliverS2 with shorter DI Restitution curve: plot APD90/DI90 relation for different BCLs

  22. Learning Luo-Rudi • Training set: for simplicity25APsgenerated from the LRd • BCL1 + DI2: from 160ms to 400 ms in 10ms intervals • Stimulus: stepwith amplitude-80A/cm2,duration0.6ms • Error margin: within 2mV of the Luo-Rudi model • Test set: 25APsfrom 165ms to 405 ms in 10ms intervals

  23. Stimulated Roadmap: One AP

  24. Stimulated Roadmap: Linear HA for One AP

  25. Stimulated Roadmap: Linear HA for One AP

  26. Stimulated Roadmap: Cycle-Linear HA for All APs

  27. Stimulated Roadmap: Cycle-Linear HA for All APs

  28. Finding Segmentation Pts Null Pts: discrete 1st Order deriv. Infl. Pts: discrete 2nd Order deriv. Seg. Pts: Null Pts and Infl. Pts Segments: between Seg. Pts Problem:too many Infl. Pts Problem:too many segments?

  29. Finding Segmentation Pts Null Pts: discrete 1st Order deriv. Infl. Pts: discrete 2nd Order deriv. Seg. Pts: Null Pts and Infl. Pts Segments: Between Seg. Pts Problem:too many Infl. Pts Problem:too many segments? • Solution: use a low-pass filter • Moving average and spline LPF: not satisfactory • Designed our own: remove pts within trains of inflection points • Solution: ignore two inflection points

  30. Finding Segmentation Pts Problem:some inflection points disappear in certain regimes Solution:ignore (based on range) additional inflection points

  31. Finding Segmentation Pts • Problem:removing points does not preserve desired accuracy • Solution: align and move up/down inflection points • - Confirmed by higher resolution samples

  32. Finding Linear HA Coefficients

  33. Finding Linear HA Coefficients

  34. Exponential Fitting • Exponential fitting: Typical strategy • Fix bi: do linear regression on ai • Fix ai: nonlin. regr. in bi ~> linear regr.in bi via Taylor exp. • Geometric requirements: curvesegments are • Convex,concave or both • Upwards or downwards • Consequences: • Solutions: might require at least two exponentials • Coefficients ai andbi: positive/negative or real/complex • Modified Prony’s method: only one that worked well

  35. Stimulated Linear HA for One AP

  36. Finding CLHA Coefficients Solution:apply mProny once again on each of the 25 points

  37. Stimulated Cycle-Linear HA for All APs

  38. Stimulated Cycle-Linear HA for All APs

  39. Frequency Response on Test Set AP on test set:still within the accepted error margin Restitution on test set:much better than we had before Frequency response: the best we know for approximate models

  40. I2 I1 b2 b1 –b2 x1 C V b1 x2 Biological Meaning of x1 and x2 Two gates: with constant conductances distributed as above

  41. Outlook: Modeling Entire Range • Modes 1&2: require 3 state variables (Na, K, Ca) • Shape changes dramatically: modes are sidestepped • Input:consider different shapes and intensities

  42. Outlook: Analysis and Control • Safety properties: • How to specify: what kind of temporal/spatial properties? • How to verify: what kind of reachability analysis? • Liveness properties: • Stability analysis:switching speed and stability/bifurcation • Controllability: • Design centralized (distributed) controllers:from CLHA • Control task: diffuse spirals and ventricular fibrillation

  43. CLHA as a TIOA

  44. CLHA as a TIOA

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