Nonlinear muscles, viscoelasticity and body taper

in the creation of curvature wavesSIAM PDEs December 10, 2007

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Tyler McMillenCalifornia State University, Fullerton

In collaboration with Thelma Williams and Philip Holmes.

Relative timing of activation and movement

Curvature travels slower than activation.

Figure from Williams, et. al., J. Exp. Biol. (1989)

How are waves of curvature created and propagated?

Why does curvature travel slower than activation?

Outline

• Elastic rod model

• Resistive fluid forces

• Discretization -- chain of rigid links

• Muscle forces

f , g - contact forces (maintain inextensibility) (Wx, Wy) - hydrodynamic body forces

Dynamics of an actuated elastic rod in the plane

Geometry and inextensibility:

Momentum balances:f

g Wy

Wx

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δ

€

κ(s − ct)

Constitutive law and free boundary conditions:

EI - bending stiffness - viscoelastic damping

actuated by time-dependent preferred curvature

Rod shape tends to its preferred shape

Curvature defines shape.

€

ϕ s →κ€

ϕ s - (actual) curvature κ - preferred curvature

Case of time-independent preferred curvature and no body forces:(no viscoelastic damping)shape oscillates around preferred shape indefinitely(viscoelastic damping positive)shape approaches preferred shape€

κ

€

δ =0

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δ > 0

Approximation of hydrodynamic forces(Following G.I. Taylor*, …to avoid doing Navier-Stokes…)

v

a

Normal forces(neglecting drag)

Normal force proportional to the product of diameter and square of velocity.

v

€

v⊥

€

v||

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N = CN ρa

Decompose forces in normal and tangential components:

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N = CN ρa + 8ρaμ L = 2.7 2ρaμ

W = Nn + Lt

- density, - viscosity

€

ρ

€

μ

*Proc. Roy. Soc. Lond. A 214, 158-183, 1952€

v⊥2

€

v⊥3 / 2

€

v⊥1/ 2v||

€

v 2

Discretization of the rod: a chain of rigid links

Use finite differences in space :

This mathematical discretizationhas a nice physical interpretationin terms of the segmented spinalcords of eels and lampreys.

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s → ih

Consider moments exerted at joints:

Force acting onjoint i:

From discretized moments:

We compute discretestiffnesses and curvatures:

Discretization: springs, dashpots and muscles

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GR ,L (t) = fR ,L (t) + νΔR ,L + γ ˙ Δ R ,L

In the continuum (small h) limit the stiffness and curvature are:

The dependence on material properties and body geometry is revealed.

Discretization: muscle properties

Stiffness and curvature are now defined in terms of body geomety, elastic properties and activation. To complete the model we need to know what the muscle forces are.

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EI ∝ν ab3, and κ ∝fR − fL

ν b

Neural activation and swimming in lamprey

The central pattern generator (CPG) of lamprey is a series of ipsi- and contralaterally coupled neural oscillators distributed along the spinal notocord. In “fictive swimming” in vitro, contralateral motoneurons burst in antiphase and there is a phase lag along the cord from head to tail corresponding to about one full wavelength, at the typical 1-2 Hz burst frequency. This has been modeled as a chain of Kuramoto type coupled rotators. The model can be justified by phase response and averaging theory:

[Cohen et al. J. Math Biol. 13, 345-369, 1982]

From Fish and Wildlife.

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GR ,L (t) = fR ,L (t) + νΔR ,L + γ ˙ Δ R ,L

At each joint model the force on either side by muscle forces. fR,L depends on:

(1) activation (calcium release, etc.) - traveling wave for now (CPG model?)

(2) length of muscle: h ± w i

(3) speed of muscle extension/contraction: ± w di/dt

CD ≈ h - w i

d(CD)/dt ≈ -w di/dt

Incorporating muscle forces

need to know this part

A model of force development in lamprey muscle

Williams, Bowtell and Curtin (*) developed a model for muscle forces based on a simple kinetic model, using data obtained from isometric and ramp experiments.

The goal of this study was to construct a model of muscle tension development which can reasonably predict the time course of muscle tension developed when muscle is stimulated at different phases during sinusoidal movement, as occurs during swimming.

The motivation for this study was to develop a model with adequate accuracy for inclusion in a full neuromechanical model of the swimming lamprey.

(*) T.L. Williams, G. Bowtell, and N.A. Curtin. Predicting force generation by lamprey muscle during applied sinusoidal movement using a simple dynamic model. J. Exp. Biol. 201:869-875 (1998)

A. Peters & B. Mackay (1961). The structure and innervation of the myotomes of the lamprey.J. Anat. 95, 575-585.

Muscle model chemical constituents:

c: calcium ions

s: calcium-binding sites in the sarcoplasmic reticulum

f: calcium-binding sites in the protein filaments

Output of CPG

d[c]/dt = k1[cs] - k2[c][s] - k3[c][f]

d[cf]/dt = k3[c][f] - k4[cf][f]

d[cs]/dt = -k1[cs] + k2[c][s]

d[f]/dt = -k3[c][f] + k4[cf][f]

d[s]/dt = k1[cs] - k2[c][s]

While the stimulus is on, k2=0.

While the stimulus is off, k1=0.

Mass action equations

Constraints

[cs] + [c] + [cf] = CT total # of calcium ions per litre is constant

[cs] + [s] = ST total # of SR binding sites per litre is constant

[cf] + [f] = FT total # of filament binding sites per litre is constant

5 equations in 5 variables plus 3 constraints 2 equations in 2 variables

Variables: [c], [cf] Parameters: k1-k5, C, S, F

k1*(CT-[c]-[cf]) Stimulus on

d[c]/dt = (k4*[cf]-k3[c])(FT-[cf]) +

k2[c](CT-ST-[c]-[cf]) Stimulus off

d[cf]/dt = (k4*[cf]-k3[c])(FT-[cf])

Reduced chemical kinetic equations

All concentration variables and parameters are made non-dimensionable by dividing by FT:

FT/FT = 1

CT/FT = C

ST/FT = S

[cf]/FT = Caf thus Caf ≤ 1

[c]/FT = Ca and Ca ≤ C

k1*(C-Ca-Caf) Stimulus on

dCa/dt = (k4*Caf-k3*Ca)(1-Caf) +

k2*Ca*(C-S-Ca-Caf) Stimulus off

dCaf/dt = (k4*Caf-k3*Ca)(1-Caf)

Chosen ad hoc

C = 2 Twice as much calcium is available than needed to bind all the filaments.

S = 6 Thrice as many binding sites are available in the SR than is required to bind all the calcium.

Scaled Chemical Equations

L = L C + LS

TC = PC (Caf, LC, VC)

TS = μS * (LS - LS0) = PC

TP = μP * L

T = PC + TP

PC = T - TP

LC(t) = L(t) - LS0 - PC(t)/μS

VC(t) = V(t) - (dPC/dt)/μS

L

LS L C

μS

μP

Mechanical model of muscle (A.V. Hill, 1938)

length-tension: force generated depends on muscle length.

Investigate using isometric experiments.

force-velocity: force generated depends on speed of lengthening or shortening of muscle.

Investigate using ramp experiments.

level of activation: force generated depends on the number of muscle fibers activated and the frequency of that activation.

Investigate by electrically stimulating muscle directly.

Muscle properties

1. The force developed is proportional to the number of calcium-activated filaments.

2. Both the length-dependence and the velocity-dependence can be described by independent multiplicative factors.

Pc = Pmax * Caf * λ(Lc) * α (Vc)

Basic assumptions of muscle model

input: desired length, velocity

l, dl/dt

measure: length, velocity

Servo motor

output: force required

a. without stimulation

b. with stimulation

Isometric experiments: constant muscle length

Ramp experiments: constant dl/dt

Sinusoidal experiments: l = lis sin (wt)

a

lis

lis and a are for a particular

preparation

stimulating electrode

Muscle experiments

Tota

l m

easu

red

forc

e -

passiv

e f

orc

e (

mN

)

λ(Lc ) = 1 + λ2(Lc-Lc0)2

P= Pmax * λ(Lc)

Isometric tetanic contractions -- length dependence

Tot

al m

easu

red

for

ce -

pas

sive

for

ce (

mN

)

αm * vc vc < 0

α (vc) = 1 +

αp * vc vc ≥ 0

PC = Pmax * Caf * λ(LC) * α (VC)

Ramp experiments -- velocity dependence

Lc(t) = L(t) - LS0 - PC(t)/μS

Vc(t) = V(t) - (dPC/dt)/μS

k1*(C-Ca-Caf) Stimulus on

dCa/dt = (k4*Caf-k3*Ca)(1-Caf) +

k2*Ca*(C-S-Ca-Caf) Stimulus off

dCaf/dt = (k4*Caf-k3*Ca)(1-Caf)

PC = Pmax * Caf * λ(LC) * α (VC)

dP/dt = k5 * (PC - P)

necessary for fitting to data

Model equations

Chosen ad hoc

C = 2 Twice as much calcium is available than needed to bind all the filaments.

S = 6 Thrice as many binding sites are available in the SR than is required to bind all the calcium.

k5=100 Chosen large enough that P closely follows Pc.

Determined from the isotonic and ramp experiments:

αm αp λ2 Pmax

Found by least-squares fit to middle-length isometric data:

k1, k2, k3, k4

Model parameters

Sinusoidal experiments & predictions

•Activation•Curvature•Rate of change of curvature

Moment Dependence

Now we have that the moment depends on:

Depends on activation and state of the rod.

Swimming

QuickTime™ and a decompressor

are needed to see this picture.

Equal activations on both sides produces “straight” line swimming

Swimming: Turns

QuickTime™ and a decompressor

are needed to see this picture.

Unequal activations on the sides produces turns.

Shapes in time

Phase lags

lamprey

simulation

It’s qualitatively correct.

Comparison of effects

What’s happening

Summary• Muscle model connected to rod “works”: it swims!• Captures qualitatively the correct behavior (phase lags, shapes,

etc.)• Model allows flexibility to explore various effects

~~~~~~~~~~~~~~~~

•More realistic fluid dynamics model (Navier-Stokes) & fluid-rod interaction (Immersed Boundary Method)•Better muscle model: Need effects of feedback and memory to get correct isometric and dynamic behavior. Connect to proprioceptive and exteroceptive sensing.•Connect models of CPG, motoneurons, muscle force interaction, fluid dynamics . . . “neurons to movement”

Future work

References• G. Bowtell & T. Williams. Anguilliform body dynamics: Modeling the interaction

between muscle activation and body curvature. Phil. Trans. Roy. Soc. B 334:385-390 (1991)

• A.H. Cohen, P. Holmes and R.H. Rand. The nature of coupling between segmental oscillators of the lamprey spinal generator for locomotion. J. Math Biol. 13:345-369 (1982)

• O. Ekeberg. A combined neuronal and mechanical model of fish swimming. Biol. Cyb. 69:363-374 (1992)

• T. McMillen and P. Holmes. An elastic rod model for anguilliform swimming. J. Math. Biol. 53:843-886 (2006)

• T. McMillen, T. Williams and P. Holmes. Nonlinear muscles, viscoelastic damping and body taper conspire to create curvature waves in the lamprey. In review, PLOS Comp. Biol.

• G.I. Taylor. Analysis of the swimming of long and narrow animals. Proc. Roy. Soc. Lond. A 214:158-183 (1952)

• T.L. Williams, G. Bowtell, and N.A. Curtin. Predicting force generation by lamprey muscle during applied sinusoidal movement using a simple dynamic model. J. Exp. Biol. 201:869-875 (1998)

• T.L. Williams, S. Grillner, V.V. Smoljaninov, P. Wallen and S. Rossignol. Locomotion in lamprey and trout: The relative timing of activation and movement. J. Exp. Biol. 143:559-566 (1989)