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MATH 680 – Final Presentation Presented by: Diem L. Bui Yu-Li Liang

“Minimal Models of bursting neurons: How multiple currents, conductance, and timescales affect bifurcation diagrams” by R. M. Ghigliazza, P. Holmes http://mae.princeton.edu/index.php?id=75. MATH 680 – Final Presentation Presented by: Diem L. Bui Yu-Li Liang. Introduction.

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MATH 680 – Final Presentation Presented by: Diem L. Bui Yu-Li Liang

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  1. “Minimal Models of bursting neurons: How multiple currents, conductance, and timescales affect bifurcation diagrams”by R. M. Ghigliazza, P. Holmeshttp://mae.princeton.edu/index.php?id=75 MATH 680 – Final Presentation Presented by: Diem L. Bui Yu-Li Liang Diem L. Bui Yu-Li Liang

  2. Introduction • Ideas of model complexity: • Generalization • The 1st dynamical Neural model was HH • Single current effects on individual cells are qualitatively understood, but collective influences have not been fully explored Diem L. Bui Yu-Li Liang

  3. Introduction • This paper: • Identification of the essential/inessential components that contribute to the bursting mechanism • Dimensional reductions • Other studies in the field Diem L. Bui Yu-Li Liang

  4. Main Finding: • A minimal model that: • Identifies biophysical parameters • Can shape and regulate key characteristics of the membrane voltage pattern • Comprise of • A 2-D fast subsystem • A very slow recovery variable, c Diem L. Bui Yu-Li Liang

  5. Presentation Overview • Review of the Hodgkin-Huxley Model • Introduction of the 3-variables Generic Model • Analysis of the effects of individual current and conductance parameters on branches of equilibria • Periodic orbits and their bifurcations • Proposal of a Minimal Model of bursting neurons • Bursting frequency • Duty Cycle • Spike Rate • Number of action potentials per burst • Conclusion Diem L. Bui Yu-Li Liang

  6. Part I: Review of the HH Model • Single Compartment ion-channel • Dynamical behaviors: • Interaction of 2 subsystems separated by time scale • Fast: governed by Sodium and Potassium • Slow: (Calcium), quasi-static behavior Diem L. Bui Yu-Li Liang

  7. Part I: Review of the HH Model (cont.) • And the subset of fast subsystem is described by the HH form: Diem L. Bui Yu-Li Liang

  8. Part I: Review of the HH Model (cont.) Diem L. Bui Yu-Li Liang

  9. Part II: 3-Variable, generic model of Bursting Neuron • Most neurons have many more membrane conductances than two • A framework that shows how biophysical parameters influence: • Existence • Stability of equilibria and periodic orbits in the fast subsystem  illuminating the global dynamics of the coupled fast-slow system (Generalization) Diem L. Bui Yu-Li Liang

  10. Part II: 3-Variable, generic model of Bursting Neuron • Class of models characterized by: • H1:Existence of a single relatively slow (non-equilibrated) variable m in the fast subsystem  homogeneous dependence on One slow variable • H2: Multiplicative dependence of conductances on gating variables, voltage, and the very slow variable c: Diem L. Bui Yu-Li Liang

  11. Part II: 3-Variable, generic model of Bursting Neuron • Using the four ions (Sodium, Potassium, chloride, and calcium) and the gating variable m: Diem L. Bui Yu-Li Liang

  12. Part II: 3-Variable, generic model of Bursting Neuron • The fast subsystem: • c varies slowly  fixed • The voltage-dependent fast and slow currents: Diem L. Bui Yu-Li Liang

  13. Part II: 3-Variable, generic model of Bursting Neuron • Fixed Points: One current • Effect of the ionic current on the location of fixed points of the fast subsytem • Separation of system into fast & slow • No influence on the location • Number of fixed points • Iss-V curves Diem L. Bui Yu-Li Liang

  14. Part II: 3-Variable, generic model of Bursting Neuron • Maximal conductance (g)  values of critical points and their location • Nernst Potential (E) fixes the unique value of voltage v=E for which the current vanishes • Threshold voltage (Vth)  affects locations and values of extrema • Slope k0 determines the extent of the transition region from the inactive state (I≈0) to the active state Diem L. Bui Yu-Li Liang

  15. Part II: 3-Variable, generic model of Bursting Neuron: I-V curve Vth<E Vth>E Diem L. Bui Yu-Li Liang

  16. Part II: 3-Variable, generic model of Bursting Neuron • Fixed Points: Multiple Currents • Linear or passive currents • Positive conductance • Destroy FP • Nonlinear currents Diem L. Bui Yu-Li Liang

  17. Part II: 3-Variable, generic model of Bursting Neuron Diem L. Bui Yu-Li Liang

  18. Part II: 3-Variable, generic model of Bursting Neuron • Bifurcation Diagrams for the fast subsystem • Fast Currents • Threshold Voltage • Slope k0 • Slow Currents • Bifurcation in term of c Diem L. Bui Yu-Li Liang

  19. Part II: 3-Variable, generic model of Bursting Neuron • Fast Currents: • Effect of Vth Vth,Ca= -38, -1.2, +15 Diem L. Bui Yu-Li Liang

  20. Part II: 3-Variable, generic model of Bursting Neuron • Fast Current: • Effect of k0 Diem L. Bui Yu-Li Liang

  21. Part II: 3-Variable, generic model of Bursting Neuron • Slow Current: • Potassium • Variation of Vth • 1st column: Iss vs. V • 2nd column: Stability • Bifurcation Diem L. Bui Yu-Li Liang

  22. Part II: 3-Variable, generic model of Bursting Neuron • Bifurcations in terms of c Diem L. Bui Yu-Li Liang

  23. Part II: 3-Variable, generic model of Bursting Neuron • The bursting mechanism Diem L. Bui Yu-Li Liang

  24. Part III: A Minimal Bursting Model • Silence, bursting, and beating • Shaping the burst Diem L. Bui Yu-Li Liang

  25. Part III: A Minimal Bursting Model • I_Ca & I_K : fast current, for oscillation • I_L: leakage current, for equilibrium point • I_KS: slow current, for bursting • I_ext: dependent on experiment need Diem L. Bui Yu-Li Liang

  26. Slow current for bursting Diem L. Bui Yu-Li Liang

  27. Part III: A Minimal Bursting ModelSilence, Bursting, and Beating • To obtain bursting, these state must coexist over some parameter range Diem L. Bui Yu-Li Liang

  28. Part III: A Minimal Bursting ModelSilence, Bursting, and Beating • Moderate increases of Iext  • (v,c) bifurcation unchanged, but shift rightward • Intersection of the nullclines to move from the lower  middle  upper branch Diem L. Bui Yu-Li Liang

  29. Part III: A Minimal Bursting ModelSilence, Bursting, and Beating • Increases in Iext: • Effects a continous change from silence  bursting  beating • Frequency increases Diem L. Bui Yu-Li Liang

  30. Part III: A Minimal Bursting ModelShaping the Bursts • 5 parameters that control the Burst: • C: spiking frequency • ε: shifting the Hopf bifurcation point to more or less depolarized levels, (global homoclinic bifurcation to left or right) • δ: recovery variable time scale  baseline bursting frequency • Iext: • influence bursting frequency, spiking frequency • Affect the number of action potential (APs) per burst • gKS: • Duty cycle (fraction of the period occupied by the burst) Diem L. Bui Yu-Li Liang

  31. Part III: A Minimal Bursting ModelShaping the Bursts • Change in Iext and gKS independently change in: • Bursting frequency • Duty cycle Diem L. Bui Yu-Li Liang

  32. Conclusion • Review of ion channel models of HH • Minimal Model is proposed: • Guidelines for creating models of specific behaviors • Select a minimal set of currents necessary produce bursting • Understand the role of biophysical parameters: • Conductance and bias currents • Bursting frequency, duty cycle, spike rate • The idea of Generalization Diem L. Bui Yu-Li Liang

  33. Merry Christmas!!Questions? Diem L. Bui Yu-Li Liang

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