Compr.Analysis of mechanical waves accompanying nerve impulse
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Compr.Analysis of mechanical waves accompanying nerve impulse
- 1. ANALYSIS OF MECHANICAL WAVES ACCOMPANYING NERVE IMPULSE Tiziano Modica SUPERVISORS: MASSIMO CUOMO, PROF. ENG. JÜRI ENGELBRECHT, PHD DSC UNIVERSITY OF CATANIA DEPARTMENT OF CIVIL ENGINEERING AND ARCHITECTURAL Master’s Degree Thesis in Civil Engineering Structural and Geotechnical CENTRE FOR NONLINEAR STUDIES, INSTITUTE OF CYBERNETICS AT TALLINN UNIVERSITY OF TECHNOLOGY, Tallinn - Estonia
- 2. Waves Monodimensional wave equation Fourier series 𝜕2 𝑢 𝜕𝑡2 − 𝑐0 2 𝜕2 𝑢 𝜕𝑥2 = 0 𝑢 = 𝐴 cos 𝑘𝑥 + 𝑐0 𝑡 = 𝐴 cos 𝑘 𝑥 + ω 𝑡 𝑘
- 3. Dispersion Non dispersive medium 𝑐 𝑝ℎ = 𝑐 𝑔𝑟 ω = 𝑐𝑜𝑠𝑡 Dispersive medium 𝑐 𝑝ℎ ≠ 𝑐 𝑔𝑟 ω = 𝑊(𝑘) Anomalous and normal dispersion
- 4. Soliton Dispersion and nonlinearity interaction Korteweg–de Vries equation Schrödinger equation Boussinesq-type equations
- 5. Pseudospectral Method (PSM) Global method Stable and accurate Boundary conditions Trigonometric functions
- 7. Gibbs phenomenon Oscillatory divergence Filtering technique
- 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 Korteweg–de Vries equation PSM approximation Solitonic behaviour
- 9. From mathematical to biological models
- 10. Nerve pulse propagation Action Potential (AP) All-or-nothing law Stable pulses Refractory period Annihilation
- 11. Cell membrane Nernst equation selective permeability -70 mV Na-K pump
- 13. Hodgkin & Huxely 1952 Potential difference Ionic currents Membrane conductance 𝑎 2𝜌𝑖Θ2 𝜕2 𝑉𝑚 𝜕𝑡2 = 𝐶 𝑚 𝜕𝑉𝑚 𝜕𝑡 + 𝑉𝑚 − 𝑉𝑁𝑎 𝐺 𝑁𝑎 + 𝑉𝑚 − 𝑉𝐾 𝐺 𝐾 + 𝑉𝑚 − 𝑉𝐿 𝐺 𝐿
- 14. Mechanical experimental evidences Heat production Swelling Curtailment
- 15. Heimburg & Jackson model 2005 1D sound propagation equation Boussinesq-type equation with solitonic solution Melting transition
- 16. Engelbrecht Peets Tamm model 2014 Nerve fibre considered as a rod Navier-Bernoulli hypothesis Rayleigh-Love correction Bishop’s model according to Porubov
- 17. Engelbrecht Peets Tamm model 2014 Inertial effects Modelling of anomalous dispersion More consistent model 𝜕2 𝑈 𝜕𝑇2 = 1 + 𝑷𝑈 + 𝑸𝑈2 𝜕2 𝑈 𝜕𝑋2 + 𝑷 + 2𝑸𝑈 𝜕𝑈 𝜕𝑇 2 − 𝑯 𝟏 𝜕4 𝑈 𝜕𝑋4 + 𝑯 𝟐 𝜕4 𝑈 𝜕𝑋2 𝜕𝑇2 Nonlinear terms Dispersive / inertial terms Solitonic initial condition Mixed derivatives
- 18. EPT model: numerical results
- 19. EPT model: numerical results H1=72.14 H2=100 P=6.58·10-4 Q=2.245·10-4 H1=72.14 H2=1 P=6.58·10-4 Q=2.245·10-4 P=Q=H1=H2=0
- 20. EPT model: numerical results H1=72.14 H2=100 P=6.58·10-4 Q=2.245·10-4 H1=72.14 H2=1 P=6.58·10-4 Q=2.245·10-4 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. 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. 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 model’s problems
- 23. Conclusions…Still unsolved Mechanical model does not represent annihilation How to Unificate HH model to EPT model?
Hinweis der Redaktion
- Phase velocity: the velocity at which the phase of any one frequency component of the wave propagates. Group velocity: is the velocity with which the envelope of the wave packet, propagates through space.
- Rayleigh-Love the transverse displacements w is associated to the longitudinal strain u proportionally to radius r and the Poisson coefficient 𝜈. Navier-Bernoulli hypothesis of planarity and normality in respect with the rod axis of plane cross sections Bishop’s model is an approximate model of wave propagation in rods suitable for the purpose since it provides, after some positions according to Porubov, the possibility to represent the anomalous dispersion (i.e. the Poisson coefficient must be negative) through the dispersion relation
- h_2 allows to take into account the inertial effects introduced by the fourth-order mixed derivatives of the Bishop’s model and, from the relative values of h_1 and h_2 is possible to model the anomalous dispersion that comes from experimental evidences J-Shaped Curve
- according to the anomalous dispersion, the shorter waves emerge in front of the signal