1. 1. ANALYSIS OF MECHANICAL WAVES ACCOMPANYING NERVE IMPULSE Tiziano Modica SUPERVISORS: MASSIMO CUOMO, PROF. ENG. JRI ENGELBRECHT, PHD DSC UNIVERSITY OF CATANIA DEPARTMENT OF CIVIL ENGINEERING AND ARCHITECTURAL Masters Degree Thesis in Civil Engineering Structural and Geotechnical CENTRE FOR NONLINEAR STUDIES, INSTITUTE OF CYBERNETICS AT TALLINN UNIVERSITY OF TECHNOLOGY, Tallinn - Estonia
2. 2. Waves Monodimensional wave equation Fourier series 2 2 0 2 2 2 = 0 = cos + 0 = cos +
3. 3. Dispersion Non dispersive medium = = Dispersive medium = () Anomalous and normal dispersion
4. 4. Soliton Dispersion and nonlinearity interaction Kortewegde Vries equation Schrdinger equation Boussinesq-type equations
5. 5. Pseudospectral Method (PSM) Global method Stable and accurate Boundary conditions Trigonometric functions
6. 6. Aliasing Cutting high frequencies Increasing the number of points
7. 7. Gibbs phenomenon Oscillatory divergence Filtering technique
8. 8. 0 10 20 30 40 50 0 20 40 60 80 100 -0.2 0 0.2 0.4 x KdV equation 3D plot T U 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Kortewegde Vries equation PSM approximation Solitonic behaviour
9. 9. From mathematical to biological models
10. 10. Nerve pulse propagation Action Potential (AP) All-or-nothing law Stable pulses Refractory period Annihilation
11. 11. Cell membrane Nernst equation selective permeability -70 mV Na-K pump
12. 12. Electrical experimental evidences Electrodiffusive models
13. 13. Hodgkin & Huxely 1952 Potential difference Ionic currents Membrane conductance 22 2 2 = + + +
14. 14. Mechanical experimental evidences Heat production Swelling Curtailment
15. 15. Heimburg & Jackson model 2005 1D sound propagation equation Boussinesq-type equation with solitonic solution Melting transition
16. 16. Engelbrecht Peets Tamm model 2014 Nerve fibre considered as a rod Navier-Bernoulli hypothesis Rayleigh-Love correction Bishops model according to Porubov
17. 17. Engelbrecht Peets Tamm model 2014 Inertial effects Modelling of anomalous dispersion More consistent model 2 2 = 1 + + 2 2 2 + + 2 2 4 4 + 4 2 2 Nonlinear terms Dispersive / inertial terms Solitonic initial condition Mixed derivatives
18. 18. EPT model: numerical results
19. 19. EPT model: numerical results H1=72.14 H2=100 P=6.5810-4 Q=2.24510-4 H1=72.14 H2=1 P=6.5810-4 Q=2.24510-4 P=Q=H1=H2=0
20. 20. EPT model: numerical results H1=72.14 H2=100 P=6.5810-4 Q=2.24510-4 H1=72.14 H2=1 P=6.5810-4 Q=2.24510-4 P=Q=H1=H2=0 0 200 400 600 800 1000 1200 1400 1600 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Plot at timestep 3900 space amplitude Full EPT equation (i.e. anomalous dispersion) EPT normal dispersion Wave equation
21. 21. Conclusions PSM is accurate and efficient, it can be used to solve PDEs and neural models Pay attention to Gibbs phenomenon and aliasing HH model is not complete, it explains only the electric behaviour of AP propagation
22. 22. Conclusions Heimburg & Jackson model explains the mechanical changes but it has some problems in terms of cph and cgr Engelbrecth Peets Tamm model solves Heimburg & Jackson models problems
23. 23. ConclusionsStill unsolved Mechanical model does not represent annihilation How to Unificate HH model to EPT model?