Morris and Lecar (1981) describe a semiqualitative model for voltage oscillations in the giant muscle fiber of the barnacle. In this system the important ions are potassium and calcium (not sodium). The equations they suggest are the following: $$ \begin{aligned} I &=C \frac{d v}{d t}+g_{\mathrm{L}}\left(v-v_{\mathrm{L}}\right)+g \mathrm{ca} M\left(v-v_{\mathrm{Ca}}\right)+g_{\mathrm{K}} N\left(v-v_{\mathrm{K}}\right) \\ \frac{d M}{d t} &=\lambda_{M}(v)\left[M_{=}(v)-M\right] \\ \frac{d N}{d t} &=\lambda_{v}(v)\left[N_{=}(v)-N\right] \end{aligned} $$ where $$ \begin{aligned} v &=\text { voltage, } \\ M &=\text { fraction of open } \mathrm{Ca}^{2+} \text { channels, } \\ N &=\text { fraction of open } \mathrm{K}^{+} \text {channels. } \end{aligned} $$ (a) Interpret these three equations. For reasons detailed in their papers Morris and Lecar define the functions \(M_{\infty}, \lambda_{s}, N_{*}\), and \(\lambda_{N}\) as follows: $$ \begin{array}{ll} M_{\infty}(v)=\frac{1}{2}\left(1+\tanh \frac{v-v_{1}}{v_{2}}\right), & N_{\infty}(v)=\frac{1}{2}\left(1+\tanh \frac{v-v_{3}}{v_{4}}\right), \\ \lambda_{M}(v)=\lambda_{M} \cosh \frac{v-v_{1}}{2 v_{2}}, & \lambda_{N}(v)=\bar{\lambda}_{N} \cosh \frac{v-v_{3}}{2 v_{4}} \end{array} $$ (b) Sketch or describe the voltage dependence of these functions. $$ \text { Note: } \begin{aligned} \cosh x &=\frac{e^{x}+e^{-x}}{2}, \quad \sinh x=\frac{e^{x}-e^{-x}}{2} \\ \tanh x &=\frac{\sinh x}{\cosh x} \end{aligned} $$ (c) Morris and Lecar consider the reduced \(v N\) system to be an approximation to the whole model. What assumption underlies this approximation? (d) Show that in the reduced \(\mathrm{vN}\) system the variables are constrained to satisfy the following inequalities: $$ \begin{aligned} 0 &

Step by step solution

01

Interpret the Equations (Part a)

The first equation describes the total ionic current and its relationship with the voltage, capacitance, leakage current, calcium current, and potassium current. The second and third equations describe the time evolution of the fractions of open calcium and potassium channels, respectively.

02

Define the Functions (Part a Continued)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Voltage Oscillations

Voltage oscillations refer to the periodic changes in voltage across the cell membrane. In biological systems like the giant muscle fiber of the barnacle, these oscillations are crucial for muscle contraction and various other cellular functions. The Morris-Lecar model provides a mathematical framework to understand these oscillations. The model uses differential equations to describe how voltage changes over time due to the movement of ions like potassium and calcium across the cell membrane. In essence, the changes in voltage are driven by ionic currents, which in turn depend on the state of ion channels and the membrane potential. This cyclic process results in oscillatory behavior, meaning that the voltage periodically increases and decreases.

Ionic Currents

Ionic currents are the flow of ions across the cell membrane, driven by electrochemical gradients. In the Morris-Lecar model, the primary ions considered are calcium (\text{Ca}^{2+}) and potassium (\text{K}^{+}). Ionic currents are represented in the equations by terms like \(g_{\text{Ca}} M(v - v_{\text{Ca}})\) for calcium and \(g_{\text{K}} N(v - v_{\text{K}})\) for potassium, where \(g\) represents the conductance, and \(M\) and \(N\) represent the fractions of open calcium and potassium channels respectively. These currents contribute to the changes in membrane potential, and their behavior can be influenced by various factors, including the voltage itself. The model assumes that these currents play a critical role in generating and sustaining voltage oscillations by repeatedly depolarizing and repolarizing the membrane.

Channel Dynamics

Channel dynamics refer to the opening and closing behavior of ion channels, which are crucial for regulating ionic currents. In the Morris-Lecar model, the fractions of open ion channels for calcium and potassium ions are given by variables \(M\) and \(N\). These fractions change over time according to the differential equations \(\frac{dM}{dt} = \lambda_{M}(v)\left[M_{\infty}(v) - M \right]\) and \(\frac{dN}{dt} = \lambda_{N}(v)\left[N_{\infty}(v) - N \right]\). The functions \(M_{\infty}(v)\) and \(N_{\infty}(v)\) represent the steady-state values of \(M\) and \(N\) as functions of voltage, while \(\lambda_{M}(v)\) and \(\lambda_{N}(v)\) are voltage-dependent rates. This dynamic behavior allows the model to simulate how changes in membrane voltage affect the state of ion channels, and hence the ionic currents, over time.

Steady State Analysis

Steady state analysis involves studying the system when it is in a stable condition, where the variables no longer change with time. In the context of the Morris-Lecar model, this means finding the values of \(v\), \(M\), and \(N\) where \(\frac{dv}{dt} = 0, \frac{dM}{dt} = 0, \frac{dN}{dt} = 0\). This analysis helps in understanding the long-term behavior of the system, rather than its transient dynamics. In steady state, the voltage and the opening fractions of calcium and potassium channels reach particular values that are maintained over time. Identifying these steady state values is crucial for predicting the behavior of biological systems under specific conditions. It also helps in understanding the system's response to perturbations, such as changes in external ionic concentrations or membrane conductance.