Possible Bifurcations
Upper Darboux sum example
Hopf Bifurcation
Lower Darboux sum example
SNIC bifurcation
Lower Darboux sum example
Homoclinic bifurcation
Current clamp simulations of the Morris-Lecar model. The injected current for the SNIC bifurcation and the homoclinic bifurcation is varied between 30 nA and 50 nA, while the current for the Hopf bifurcation is varied between 80nA and 100nA

Kinetics

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There are many different mathematical descriptions of ion channels but arguably the most accurate is the single channel kinetics using Markov Chains. An ion channel can exist in many different conductance states with transition rates governed by the voltage across the cell membrane.

Let us begin with a simple hypothetical Ion channel with two states, open   and closed  . The closed state transitions to the open state at a rate   while the open state transitions to the closed state at rate  . Note that both   and   are functions of voltage  .

 

Following Michaelis–Menten kinetics we can write down a formula for how a population of ion channels behave. Let   denote the percentage of ion channels in the open state. Similarly   is the percentage of ion channels in the closed state.

 

Conveniently   and  .   is known as the steady state and   is the time constant.

The most usual function for the stead state activation is usually a Boltzmann equation given by the form

 

Random Equations

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