Action potentials and the Hodgkin-Huxley formalism

Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,Action potentials and the Hodgkin-Huxley formalism,3-4 nM,1(,M),2 of,cytoplasm has 10,10,water molecules and 10,8,ions.,In water,the molecules Na,+,K,+,Cl,-,and Ca,2+,are in ionic form,The voltage on the inside of a neuron in typically-50 to-90mV lower than in the extracellular fluid.,V,1,n,1,V,2,n,2,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,+,Concentration gradient,Voltage gradient,Nernst equation:,(inside)(outside),A derivation of this can be seen in Johnston and Wu,pg10-14,Nernst equation:,Depending on the temperature:,Examples:,Sodium ions:n,1,=60 mM n,2,=440mM E=27,ln,(440/60)=54mV,Potassium ions:n1=400 mM n2=20 mM E=27,ln,(20/400)=-80mV,Chloride ions:E=-65mV,Calcium ions:E=130mV,Current through an ion channels,For one ion type with reversal potential E:,Basically,this is ohms law,where g is the conductance;the inverse resistance(R).,For several ions through different channels:,The neuron as an RC circuit,C,m,R,m,=1/g,m,V,I,I,c,I,m,where,+,-,Therefore:,The neuron as an RC circuit,C,m,R,m,V,I,I,c,I,m,The equation:,has the solution:,IR,m,C,m,g,i,E,i,For a channel with a reversal potentianl,E,i,actually have:,With the equation:,V,I,I,c,I,m,For several ionic channels,with different reversal potentials:,If a Neuron is an RC circuit how can it spike?,Voltage dependant conductances:,Simple case,an,activating current:,Interpretation:n the probability that a channel is open.,n,V(mV),open closed,n (1-n),open closed,n (1-n),Where:,and,We have seen similar equations.For a fixed V,the equations converge exponentially to ,with a time constant,The potassium channel behaves in this way,but HH assume that:P,k,=n,4,Inactivating channels sodium:,m activation,h inactivation.,HH assume that:P,na,=m,3,h,The same type of differenttial,equations can be set up for m and h.,The complete HH equation,for the generation of action potential takes the form:,HH action potential,potential,channel activation,HH action potential,Assignment 2:(due Feb 13th),Program in matlab a Hodgkin-Huxley neuron with parameters from,DA-page 172.What happens if the Potassium conductance is set to 0?,Explore different input currents for different durations.,Is there a threshold voltage at which an AP is induced.,Program a Connor-Stevens model,DA-page 196,compare to the HH model.,In an HH model,what happens when the K conductance is set to zero?,It is best not to use an ODE solver.,Here it will be sufficient to use a first order method such as the forward Euler method,Possible texts for numerical ODE solutions:,Appendix A of Chapter 5 in the DA book,Chapter 14 in the book Methods in Neuronal modeling by Koch and,Segev,second edition.Chapter written by,Mascagni,and Sherman),Simplifying HH:,The HH equation is a 4 dimensional equations(V,m,h,n),how can it be simplified?,Note,m,is much smaller than,n,h,therefore set:,Simplifying HH:,This produces a reduced 3D model(V,h,n),how close is it to the HH equation?,Can we reduce dimensionality further?,Assume:h=0.9-n,get a 2D model(V,n)note that,n,and,h,have similar time constants.,What kind of AP do we get?,How,far can we go?,What happens if we set:,1D model:,Too simple!,2D model,(1),(2),2D model,(1),(2),General form:,where,R=1/g,L,and,=RC,m,A simple example of fixed points and stability in 1D:,It is hard to solve such a non-linear equation.Instead of solving we use other methods.,Fixed points,when:,Therefore:,General form:,FitzHugh-Nagumo model:,F,(,u,w,),=,u,-,u,3,-,w,G,(,u,w,),=,b,0,+,b,1,u,-,w,Phase plane analysis:,Nullclines:the two lines and,Fixed points,the intersection of nullclines.,For FN model:,Direction of flow plot,Phase plane analysis:,Direction of flow:,All intersection are locally linear lets look at linear case,Stability of fixed points 1D example:,Fixed points:,Define:,In general:,For this example,and for have:,Stability of fixed points 1D example:,Fixed points:,Define:,For this example,and for have:,We have seen such differential equations today,what is their solution?,Stability in 2D:(simplified expression),define:,Formally:,eigen-values,eigen-vectors?,But what does it mean?,Eigen value equation:,For a 2D problem get two eigen-values:,And two eigen-vectors:,The solution to the differential equation:,If both the fixed point is stable.,The matrix:is a rotation matrix,such that:,This is called diagonalizing the matrix.,We have that:,and that:,Therefore necessary and sufficient conditions for stability are,Linear example:,nullclines:,