The FitzHugh-Nagumo Model

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1 Lecture 6 The FitzHugh-Nagumo Model 6.1 The Nature of Excitable Cell Models Classically, it as knon that the cell membrane carries a potential across the inner and outer surfaces, hence a basic model for a cell membrane is that of a capacitor and resistor in parallel. The model equation takes the form dv C m = V V eq + I R appl, (6.1) here C m is the membrane capacitance, R the resistance, V eq the rest potential, V the potential across the inner and outer surfaces, and I appl represents the applied current. In landmark patch clamp experiments in the early part of the 2 th century, it as determined that many cell membranes are excitable, i.e., exhibit large excursions in potential if the applied current is su ciently large. Examples include nere cells and certain muscle cells, e.g., cardiac cells. >From , Hodgkin and Huxley conducted patch clamp experiments on the giant squid axon, a rather large part of nere tissue suitable for experimentation gien the technology of the time. Based on their experiments, they constructed a model for the patch clamp experiment in an attempt to gie mathematical explanation for the axon s excitable nature. A key part of their model assumptions as that the membrane contains channels for potassium and sodium ion o. In e ect, the 1/R factor in (6.1) became potential dependent for both channels. The underlying model equation is: C m dv = g Kn 4 (V V K ) g Na m 3 h(v V Na ) g L (V V L )+I appl. (6.2) Here the subscripts K, Na, and L correspond to potassium, sodium, and leakage channels, respectiely. The terms g K n 4, g Na m 3 h, and g L are the conductances 55

capacitor and resistor in parallel. The model equation takes the form dv C m = V V eq + I R appl, (6.

2 (reciprocal of resistances). The ariables n, m, and h are hypothesized potential dependent gating ariables hose dynamics ere assumed to follo rst order kinetics. The equations take the form τ (V) d = 1(V), =n, m, h, (6.3) here τ (V) and 1 (V) are the time constant and rate constant determined from the experimental data. Taken together, (6.2) and (6.3) represent a four dimensional dynamical system knon as the Hodgkin-Huxley model. It does proide a basis for qualitatie explanation of the formation of action potentials in the giant squid axon. Moreoer, the model structure forms a basis for irtually all models of excitable membrane behaiour. 6.2 FitzHugh Model Reduction In the mid-195 s, FitzHugh sought to reduce the Hodgkin-Huxley model to a to ariable model for hich phase plane analysis applies. His general obserationas that the gating ariables n and h hae slo kinetics relatie to m. Moreoer, for the parameter alues speci ed by Hodgkin and Huxley, n + h is approximately.8. This led to a to ariable model, called the fast-slo phase plane model, of the form C m dv = g Kn 4 (V V K ) g Na m 3 1 (V)(.8 n)(v V Na) g L (V V L )+ I appl n (V) dn = n 1(V) n. Ine ect this proides a phase space qualitatie explanationof the formation and decay of the action potential. (See Keener & Sneyd 1.) A further obseration due to FitzHugh as that the V -nullcline had the shape of a cubic function and the n-nullcline could be approximated by a straight line, both ithin the physiological range of the ariables. This suggested a polynomial model reduction of the form d d = ( α)(1 ) +I = ε( γ). (6.4) 1 J. Keener & J. Sneyd, Mathematical Physiology, Springer-Verlag (1998), p

3) represent a four dimensional dynamical system knon as the Hodgkin-Huxley model. It does proide a basis for qualitatie explanation of the formation of action potentials in the giant squid axon.

3 Here, the model has been put in dimensionless form, represents the fast ariable (potential), represents the slo ariable (sodium gating ariable), α, γ, and ε are constants ith < α < 1 and ε 1 (accounting for the slo kinetics of the sodium channel). In 1964, Nagumo constructed a circuit using tunnel diodes for the nonlinear element (channel) hose model equations are those of FitzHugh (6.4). Hence the equations (6.4) hae become knon as the FitzHugh-Nagumo model. 6.3 Simulations and Phase Plane We assume the parameters are such that precisely one equilibrium point exists. Further, e translate the model to place this equilibrium at(,) as follos. Let f()= ( α)(1 ), and let( eq, eq ) be the equilibrium point for (6.4). Then e can rite the model equations in the form d d = f(+ eq ) f( eq ) = ε( γ). Here e ill illustrate the behaiour of the FitzHugh-Nagumo model using typical alues for the parameters. To Hopf bifurcation phenomena ill be illustrated by arying eq and by arying α. In the rst case, e take alues: α=.139, ε=.8, γ=2.54. The plots are shon in Figures 6.1 and 6.2 here e hae set eq =.7 and=.15, respectiely. The phase portraits ith nullclines are shon on the left. Note ho the orbits are drien by the nullclines. Further, note the position of the knee of the -nullcline in relation to the equilibrium point in the to gures. (In the limit cycle case, the knee is to the left.) Numerically, the actual bifurcation alue is approximately eq =

Hence the equations (6.4) hae become knon as the FitzHugh-Nagumo model. 6.3 Simulations and Phase Plane We assume the parameters are such that precisely one equilibrium point exists.

4 t Figure 6.1: FitzHugh-Nagumo: eq = t Figure 6.2: FitzHugh-Nagumo: eq =.15 In our second case e take eq =and allo α to ary and take on negatie alues. The plots are shon in Figures 6.3 and 6.4 here e hae set α=.139 and =.139, respectiely. The other parameters are ε =.8 and γ = 1.5. Again e note in the phase plots ho the orbits follo the nullcline elds and the locationof the knee of the -nullcline for the limit cycle. 58

3 and 6.4 here e hae set α=.139 and =.139, respectiely. The other parameters are ε =.8 and γ = 1.5.

5 Figure 6.3: FitzHugh-Nagumo: α= t Figure 6.4: FitzHugh-Nagumo: α= Bifurcation Analysis Here e ill present the bifurcation analysis corresponding to Figures 6.3 and 6.4. We ill see that the bifurcation point is α = εγ ith a limit cycle bifurcating for α < α. Computing the Jacobian e hae α 1 J=J(,)=. ε εγ 59

4 Bifurcation Analysis Here e ill present the bifurcation analysis corresponding to Figures 6.3 and 6.4. We ill see that the bifurcation point is α = εγ ith a limit cycle bifurcating for α < α.

6 Hence, Tr(J)= (α+ εγ) anddet(j)= ε(αγ+1). The eigenalues are gien by λ= (α+εγ) p (α+εγ) 2 4ε(αγ+1), 2 and the condition for the eigenalues to be complex reads: εγ 2ε 1/2 < α < εγ+2ε 1/2. At α = εγ, the eigenalues are λ = iη, here η = p ε(1 εγ 2 ). The corresponding eigenectors are gien by: 1 1 and. εγ iη εγ+ iη We can transform the model equations into suitable form using the transformation 1 T =, ith T εγ η 1 = 1 η. η εγ 1 Setting = T x y, e obtain _x _y = µ η εγµ η + η x y " (1 (µ+εγ)x 2 x 3 + εγ(1 (µ+εγ)x 2 +εγx 3 η #, (6.5) here e hae set µ= (α+εγ). This is the form appropriate to apply Theorem 5.4. We hae d= d dµ Reλ = 1 α= εγ 2 >. Further, denoting the nonlinear terms in (6.5) by f(x, y) and g(x, y), then a = 1 16 f xxx+ 1 16η ( f xxg xx ) = γ (1 εγ) εγ 2. (,, εγ) For the parameters gien aboe (ε=.8, γ =1.5) e hae a=.2.. <. Consequently e can conclude that a limit cycle bifurcates for µ >, i.e. α < εγ. This is hat as desired. 6

εγ iη εγ+ iη We can transform the model equations into suitable form using the transformation 1 T =, ith T εγ η 1 = 1 η.

Simulation of an Action Potential using the Hodgkin-Huxley Model in Python. Nathan Law 250560559. Medical Biophysics 3970

Simulation of an Action Potential using the Hodgkin-Huxley Model in Python Nathan Law 250560559 Medical Biophysics 3970 Instructor: Dr. Ian MacDonald TA: Nathaniel Hayward Project Supervisor: Dr. Andrea