Modeling the muscular response to motor neuron spike-trains. Laura Miller and Katie Newhall SAMSI Transition Workshop May 4, 2016

Transcription

1 Modeling the muscular response to motor neuron spike-trains Laura Miller and Katie Newhall SAMSI Transition Workshop May 4, 2016

2 Outline 1. Motivation for an integrative neural and mechanical view of animal locomotion and behavior. 2. Preview of the specific examples we considered. 3. Background on the biology and mathematical modeling of muscle. 4. How motor neuron spike trains result in muscle force generation: Biology and modeling. 5. Improvements to an integrative mathematical model.

3 Beyond Brains Call for integrated view of neural and biomechanical basis of behavior - Michael Dickinson, 2010 Allen Institute for Brain Science Symposium.

4 Applications Prosthetics and robots Goals: To understand how people adapt to robotic exoskeletons and bionic prostheses with the intent of improving technologies for rehabilitation Restore lost function for disabled patients by designing prosthetic devices that can translate neural brain activity

5 Applications

6 Applications and Data Available Gait analysis and pathological gaits

7 An Integrated View of Neuromechanics Example: Fish Swimming

8 Integrated Model from Tytell, Williams, et al. Credit: Eric D. Tytell, Tufts University

9 From Neuron Signals to Locomotion How does a jellyfish translate a neural signal into a swimming mode? How do phenomena such as muscle force-length relations and muscle stiffness affect swimming performance? How does a jellyfish adjust its behavior to maneuver or turn?

10 From Neuron Signals to Locomotion Model of insect walking Goal: Replace applied external force function in peg leg model with a model of the muscle. Future: Can we then integrate motor neuron activity to control walking? Philip Holmes, Robert J. Full, Dan Koditschek, and John Guckenheimer, SIAM Review, 2006

11 Muscle Activation Motor neuron signal - spike train. Action potentials along muscle cell. Release of Calcium ions from the SR. Calcium binding to troponin - regulates crossbridge formation. Myosin heads complete crossbridge cycle, myosin slides past actin. Muscle shortens and force is generated.

12 Spike Train to Force Dynamics In tetanic (full) contraction, spike trains are fast enough that the muscle does not relax between contractions. At lower frequencies, the muscle will relax between successive stimuli, and the tension produced will vary in time.

13 Sliding Filament Model: Muscle Contraction Thick filaments (myosin) slide across thin filaments (actin). Myosin heads burn ATP while forming crossbridges that pull on the actin. The instantaneous force of contraction depends on the instantaneous velocity. The force of contraction also depends on the current length of the muscle.

14 Hill Model: Force-Velocity Relationship The Hill muscle model describes the force-velocity relationship is for tetanic (fully activated) contraction. This is a steady state model in which the constants, a and b, are experimentally derived. Vmax = maximum velocity under no load P0 = Maximum load with no contraction Constants can be related to crossbridge dynamics.

15 Length-Tension Relationship At intermediate lengths, all of the myosin heads are within reach of the thin filaments. When stretched, few myosin heads overlap with the thin filaments. When fully contracted, interference reduces force.

16 Force-Velocity-Length Model Assume that the force produced for a given length, LF, and velocity may, VF, be found by multiplying the normalized Hill and Force-Velocity models. First rearrange terms and normalize: Force-Length Model Force-Velocity Model

17 2-Element Hill Muscle Model CE - contractile element SE - series element (nonlinear spring) lc - length contractile element vc - velocity contractile element Force generated by muscle is a function of activation strength, current length, lc, and current contractile velocity, vc. Contractile element is attached to a spring representing the tendon and connective tissues (resists long extensions).

18 Calcium Dynamics Model uses law of mass action to model the rates that calcium ions are bound and released from the actin filaments and the sarcoplasmic reticulum. Sarcoplasmic Reticulum (SR) Actin Filament

19 Williams Model [c] Free Calcium [cs] Calcium bound to SR [s] Free binding sites in SR [cf] Calcium bound to Filament [f] Free binding sites on Filament Williams, T.L. J Exp Bio 213, (2010)

20 Williams Model Reduction Total Calcium Ions Total SR binding sites Total filament binding sites 5 Unknowns, 3 Constraints, Reduce to 2 variables: Stimulus ON Stimulus OFF

21 Our Modification to Williams Model (1) (2) (3) Caf unbinding only proportional to number bound calcium Ca uptake to SR always present Release rate of calcium from SR function of time to model spike activation

22 Bifurcation Diagram Modification changes fixed points, choose new C, S parameters (1) Want (0,0) stable when no stimulus (2) Want Caf close to 1 when stimulus on At fixed C=10 Want S>C, Caf decreases with increasing S For C=S Choose S=C=200

23 Connect to Hill Model force Control strength of activation force-length force-velocity

24 Spike-Activation Models Modify k1(sr release rate) to model spike-activated release of Calcium from SR square-wave square-pulse a - height nu - frequency a - height tau - time on nu - frequency spike-pulse b - jump nu - Spike-time frequency tau_d - decay time constant

25 Spike-activated Model Simulation 5Hz spike-pulse 50Hz spike-pulse

26 Maximum Force vs. Frequency Biologically, expect force to increase with frequency Frequency

27 Summary - Important to develop integrated models for the neuromuscular junction We adapted a model to study spike-activation Found biologically realistic force vs. frequency relationship for some models Future Directions: - Match to experimental data (invertebrates) - Spike-time input into model, match force as function of time - Couple to stick-leg model of locomotion

28 Acknowledgements The authors would like to thank the organizers of the SAMSI Program on Computational Neuroscience. A special thank you also goes to Thomas Witelski for his help planning our workshop as well as the entire program.

29 Length-Tension Model FI - maximum isometric tension at a given length FIO - maximum isometric force produced at the optimum length of the muscle fibers LF - length of the muscle fibers LFO - length at which the muscle fibers exert their maximum tension SK is a constant specific for each muscle where SK > 0.28.