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This paper explores minimal models of bursting neurons, focusing on how various currents, conductance values, and timescales influence bifurcation diagrams. The study aims to identify crucial components shaping bursting mechanisms and proposes a 2-D fast subsystem combined with a slow recovery variable to regulate membrane voltage patterns. By analyzing individual current and conductance parameters, the research sheds light on bursting frequency, duty cycle, spike rate, and action potential numbers per burst. Overall, the findings offer insights into the dynamics of bursting neurons and the role of key biophysical parameters.
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“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
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
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
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
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
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
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
Part I: Review of the HH Model (cont.) Diem L. Bui Yu-Li Liang
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
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
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
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
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
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
Part II: 3-Variable, generic model of Bursting Neuron: I-V curve Vth<E Vth>E Diem L. Bui Yu-Li Liang
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
Part II: 3-Variable, generic model of Bursting Neuron Diem L. Bui Yu-Li Liang
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
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
Part II: 3-Variable, generic model of Bursting Neuron • Fast Current: • Effect of k0 Diem L. Bui Yu-Li Liang
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
Part II: 3-Variable, generic model of Bursting Neuron • Bifurcations in terms of c Diem L. Bui Yu-Li Liang
Part II: 3-Variable, generic model of Bursting Neuron • The bursting mechanism Diem L. Bui Yu-Li Liang
Part III: A Minimal Bursting Model • Silence, bursting, and beating • Shaping the burst Diem L. Bui Yu-Li Liang
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
Slow current for bursting Diem L. Bui Yu-Li Liang
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
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
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
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
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
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
Merry Christmas!!Questions? Diem L. Bui Yu-Li Liang