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MBW:The Morris-Lecar Model for Excitable Systems

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This is a summary of the paper Morris, C. and Lecar, H. (1981) Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35: 193 - 213.


Overview

The biological system studied was the Barnacle muscle through its membrane potential and the concentrations which had an effect on it. This model used a simplified version of the Hodgkin-Huxley model. A reduced system of equations was created that produce the same results for most cases. In order to approximate the muscle reactions appropriately, the models are based off of one you would use to follow an equivalent circuit design. For more on the Hodgkin-Huxley model, please visit MBW:Hodgkin-Huxley.

Introduction

In 1981, Morris and Lecar proposed their eponymous equation as an reduced excitation model in the vein of the Hogdkin-Huxley equations, which constitutes an exact description of the conductance-based model for the giant squid neuron. In particular, they begin with an "already reduced" system of equations taken from the voltage clamp studies of Keynes, et. all [1], which proposes that the relevant state variables for the barnacle muscle are, along with the membrane potential, the proportion of voltage dependent Ca+ and K+ ion channels open at a given time (which is to say, the Na+ conduction plays no significant role in dynamics):


I=C{\dot  {V}}+g_{L}(V_{L})+g_{{Ca}}M(V-V_{{Ca}})+g_{K}N(V-V_{k})

{\dot  {M}}=\lambda _{M}(V)(M_{{\infty }}(V)-M)

{\dot  {N}}=\lambda _{N}(V)(N_{{\infty }}(V)-N)


Which has the circuit equivalent:

Lecar circuit.png

From here, they make various assumptions about independence (or uncoupling of their dynamics) of the Ca++ and K+ channels, as well as assumptions about their relative recovery rates in order to further reduce the dimension of the model while preserving the essential character of the system's long term behavior.

While this reduced model exhibits a rich variety of behaviors, not all of the possible behavior observed in excitable systems, such as bursting and chaos, can described it.

Parameter list and A Qualitative Discussion of the Non-reduced model

Parameter List

I= applied current (\mu A/cm2)

I_{L},I_{C},I_{K} = leak, Ca++, and K+ currents, respectively (\mu A/cm2)

g_{L}.g_{{ca}}.g_{K}= maximum or instantaneous conductance values for leak, Ca++, and K+ pathways,respectively (mmho/cm2)

g_{C} = conductance constant for nonlinear I_{{ca}} (mmho/cm2)

V = membrane potential (mV)

V_{L},V_{{Ca}},V_{K} = equilibrium potential corresponding to leak, Ca++, and K+ conductances, respectively (mV)

M = fraction of open Ca++ channels

N = fraction of open K+ channels

M_{{\infty }}(V),N_{{\infty }}(V) = fraction of open Ca++ and K+ channels, at steady state

\lambda _{M}(V),\lambda _{N}(V) = rate constant for opening of Ca++ and K+ channels

{\bar  {\lambda }}_{M},{\bar  {\lambda }}_{N} = maximum rate constants for Ca++ and K+ channel opening

V_{I}= potential at which M_{{\infty }}=0.5(mV)

V_{2} = reciprocal of slope of voltage dependence of Mz. (mV)

V_{3} = potential at which N_{{\infty }}=0.5(mV)

V_{4} = reciprocal of slope of voltage dependence of NCO (mV)

C = membrane capacitance (\mu F/cm2)

where

M_{{\infty }}(V)=1/2(1+\tanh((V-V1)/V2))

\lambda _{M}(V)={\bar  {\lambda _{M}}}\cosh((V-V1)/2V2)

N_{{\infty }}(V)=1/2(l+\tanh((V-V3)/V4)

\lambda _{N}(V)={\bar  {\lambda }}_{N}\cosh((V-V3)/2V4).

Discussion of non-reduced model

In order to see how equations (1) might be an intuitive model for voltage dependence ion channel, fix V. Consider the 2nd (1); if M>M_{{\infty }}(V), then we have that {\dot  {M}}<0, so that the solution M(t) is decreasing. On the other hand, if M<M_{{\infty }}(V), then we have that {\dot  {M}}>0, so that the solution M(t) is increasing. Thus, M=M_{{\infty }} is a stable equilibrium as expected. A similar sequence of ideas applies to N=N_{{\infty }}, and we can conclude that in absence of voltage fluctuation, the number Ca++ and K+ ion channel tends to a constant determined by its Boltzmann distribution (M_{{\infty }}(V),N_{{\infty }}(V)) for the potential.


Membrane potential.png

Now consider non-constant voltage; in light, of above we can think of N and M "following" the moving voltage. However, since the voltage itself depends on the number of open Ca++ and K+ channels, this ODE describes a feed back loop whereby the voltage gradient increases as if the net number of open ion pathways is decreasing, while an increasing voltage forces more ion channels to open and produces more current which depletes the voltage once V becomes greater than V_{{Ca}} or V_{K}, creating a kind of push-pull, ebb and flow mechanism. It is easy to see that this push-pull mechanism is partially dependent (in addition to the the applied current) on the values of the Calcium and Potassium conduction, g_{{Ca}} and g_{K}. This along with the fact that the Ca++ K+ have opposite signs in the equation for voltage, suggests: given a large enough ratio of g_{{Ca}} relative to g_{K} (and with an adequate amount of applied current) sustained or dampened oscillations might occur. Indeed this case, as we shall see in Existence of Limit Cycles section.

The Experiment

Morris and Lecar employed a technique known as current clamping and space clamping, which essentially eliminated spatial variations the traveling action potential and fixed the current across the membrane to set parameter which the experimenter could control [2].

Then in order to reduce the model, they submerged the barnacle muscle in various solutions in order to eliminate either Ca+ and Na+ conductance, or in the case of the Morris-Lecar equation, simultaneously optimize them. By doing this, they reduced the dynamics of the system to two state variables (depending on which solution the barnacle is submersed in), which considerably simplifies possible behavior the system can exhibit due to special topological properties of the phase plane that is not enjoy by higher dimensional Euclidean space (namely that a simple closed curve divides the plane into two distinct connected components. See [1] [2].

Pure K+ conductance

When the barnacle is submersed in a Ca+ free solution, Ca+ conductance is eliminates and eq (1) reduces to:


I=C{\dot  {V}}+g_{L}(V_{L})+g_{K}N(V-V_{k})

{\dot  {N}}=\lambda _{N}(V)(N_{{\infty }}(V)-N)

This system has precisely one fixed-point which is always an attractor. Here is a typical orbit for this system:

Lecar1.gif

Pure Ca+ condunctance

Likewise, when the barnacle is submersed in the solution that minimizes the K+ conductance, we have

I=C{\dot  {V}}+g_{L}(V_{L})+g_{{Ca}}M(V-V_{{Ca}})

{\dot  {M}}=\lambda _{N}(V)(M_{{\infty }}(V)-M)

The situation for pure Ca+ conductance is more complicated. Here, in addition to an attracting fixed-point, the fixed point can be a saddle point and a second stable fixed point appears. See the section Bifurcations for details.

Morris-Lecar equation

The main idea behind the derivation of the Morris-Lecar equation is that relative to the K+ ion channel, the recovery of Ca++ state variable is instaneous, i.e. M(t)=M_{{\infty }}(V(t)). The we obtain


I=C{\dot  {V}}+g_{L}(V_{L})+g_{{Ca}}M_{{\infty }}(V)(V-V_{{Ca}})+g_{K}N(V-V_{k})

{\dot  {N}}=\lambda _{N}(V)(N_{{\infty }}(V)-N)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (4)

Existence of Limit Cycles

Morris and Lecar prove the existence of a limit cycle for certain values of the applied current and the ion conductances g_{{Ca}} and g_{{K}}, in the cannonical way. That is, they construct a trapping region which contains no fixed point, and then apply the Poincare-Bendixson theorem. Their argument is reproduced below:

First, we note that, intuitively, this physical system must have finite energy, and therefore is bounded in some sense. To see this, we note the state variable N, can only takes on value between 0 and 1, and thus remains in this interval. Likewise, given a fixed applied current, the analogous circuit cannot exceed the total energy within it, and is bounded by

V_{{min}}={\frac  {g_{L}V_{L}+g_{K}V_{K}+I}{g_{L}+g_{K}}}<V<{\frac  {g_{L}V_{L}+g_{{Ca}}V_{{Ca}}+I}{g_{L}+g_{{Ca}}}}

which is derived from the fact that when V={\frac  {g_{L}V_{L}+g_{K}V_{K}+I}{g_{L}+g_{K}}} or {\frac  {g_{L}V_{L}+g_{{Ca}}V_{{Ca}}+I}{g_{L}+g_{{Ca}}}},{\dot  {V}}=0, and thus must be bounded by these two limiting potentials.

Thus we have shown that any orbit with the rectangle T=[{\frac  {g_{L}V_{L}+g_{K}V_{K}+I}{g_{L}+g_{K}}},{\frac  {g_{L}V_{L}+g_{{Ca}}V_{{Ca}}+I}{g_{L}+g_{{Ca}}}}]\times [0,1] must remain in T.

Second, we linearize system (4) by finding its Jacobian matrix of partial derivatives and evaluate it at the fixed point (V^{*},N^{*}) (which is contained in the rectangle T, as shown by the nullclines ). From here, we obtain the characteristic polynomial, the solutions of which are the eigenvalues of the linearized ODE:

p^{2}-(({\frac  {\partial f_{1}}{\partial V}}+{\frac  {\partial f_{2}}{\partial N}})_{{(V*,N^{*})}})p+({\frac  {\partial f_{1}}{\partial V}}{\frac  {\partial f_{2}}{\partial N}}+{\frac  {\partial f_{2}}{\partial V}}{\frac  {\partial f_{1}}{\partial N}})_{{(V*,N^{*})}}=0.

In order to have a stable limit cycle, both roots must be positive if they are real or have a positive real part if they are complex. Thus we could apply the Hartman-Grobman theorem, and conclude that the Morris-Lecar equation is conjugate (call it h) to the unstable linearization. Thus, say if (4) is conjugate in a \epsilon neighborhood of the equilibrium, then any initial condition (for the linearized system) on the closed curve B=\{(x,y):x^{2}+y^{2}=(e/2)^{2}\} can never enter the region R=\{(x,y):x^{2}+y^{2}<(e/2)^{2}\}. Therefore, h(B) forms the inner boundary of our trapping region for equation 4, and hence created a trapping region which contains no fixed points. Thus we can conclude by Poincare-Bendixson that orbits with in this region must approach a periodic orbit.

In order to guarantee, that the real parts of the eigenvalues of the linearized system are positive, we must have that:

({\frac  {\partial f_{1}}{\partial V}} + {\frac  {\partial f_{2}}{\partial N}})_{{V*,N^{*}}}>0

({\frac  {\partial f_{1}}{\partial V}}{\frac  {\partial f_{2}}{\partial N}}+{\frac  {\partial f_{2}}{\partial V}}{\frac  {\partial f_{1}}{\partial N}})_{{(V*,N^{*})}}>0.

Substituting equation (4) into the above we get

g_{{ca}}{\frac  {\partial M_{{\infty }}}{\partial V}}(V_{{ca}}-V^{*})>g_{L}+g_{K}N^{*}+g_{{Ca}}M_{{\infty }}-(V^{*})+C\lambda _{N}(V^{*}),


and

g_{{Ca}}{\frac  {\partial M_{{\infty }}}{\partial V}}(V_{{Ca}}-V^{*})<g_{L}+g_{K}N^{*}+g_{{Ca}}M_{{\infty }}(V^{*})+g_{K}{\frac  {\partial N_{{\infty }}}{\partial V}}(V^{*}-V_{K})

which gives the sufficient conditions, and defines a parameter space, under which limit cycles occur.

Bifurcations

For fixed g_{{Ca}},g_{{K}}, as the applied voltage increased past certain thresholds, the Morris-Lecar equation can under go a series of bifurcation which exhibits the entire range of behavior discussed above:

Lecar2.gif


In particular, starting at I=0mA and increasing the applied current density to 110mA/cm^{2}, this instance of the Morris-Equation undergoes a saddle-node bifurcation and a Hopf bifurcation.

Also, for a fixed applied current, we can construct a bifurcation diagram in terms of the ion conductance g_{{Ca}} and g_{{K}}:

Bifurcation param.png

Here we can see a transition from the stable node behavior and the bi-stable saddle

Lecar4.gif

Notice that as G_{{Ca}} increases beyond the bifurcation point (for G_{K}=8). two stable equilibrium appear, each basin of which having a common boundary of the separatrix formed by the stable manifold of the saddle point equilibrium.

Conclusion

The Morris-Lecar model for excitable systems is a simplified form of the Hodkins-Huxley equations that captures much of the behavior of the higher-order system. By reducing the system to two state variables, analysis become more tenable and the theory of bifurcations in the plane can applied in order to rigorously show the existence of mono- and bi-stable orbits, Limit cycles, and dampened oscillations for certain parameters.

See also, The Behavior of Traveling Waves in excitable media

Citations

[1] Keynes, R. D., E. Rojas, R. E. Taylor, and J. Vergara. 1973. Calcium and potassium systems of a giant barnacle muscle fibre under membrane potential control. J. Physiol. (Lond.) 229:409-455.

[2] N. Britton, Essential Mathematical Biology. (2003)