*Understanding the Brain*

# Morris–Lecar model

The **Morris–Lecar model** is a biological neuron model developed by Catherine Morris and Harold Lecar to reproduce the variety of oscillatory behavior in relation to Ca++ and K+ conductance in the giant barnacle muscle fiber.^{[1]} Morris-Lecar neurons exhibit both class I and class II neuron excitability.

## Contents

## Experimental method

The Morris-Lecar experiments relied on the current clamp method established by Keynes et al. (1973).^{[2]}

"Large specimens of the barnacle Balanus nubilus (Pacific Bio-Marine Laboratories Inc., Venice, Calif.)
were used. The barnacle was sawed into lateral halves, and the depressor scutorum rostralis muscles
were carefully exposed. Individual fibers were dissected, the incision starting from the tendon. The other
end of the muscle was cut close to its attachment on the shell and ligatured. Isolated fibers were either
used immediately or kept for up to 30 min in standard artificial seawater (ASW; see below) before use.
Experiments were carried out at room temperature of -22 C." ^{[1]}

### Assumptions

- Equations apply to an iso-potential patch of membrane.
- Ca++ carries the depolarizing current, K+ carries the hyperpolarizing current.
- Activating conductance quickly relaxes to its steady state value independent of voltage.
- "The dynamics of the recovery variable can be approximated by a first-order linear differential equation for the probability of channel opening. This assumption is never exactly true, since channel proteins are composed of subunits, which must act in concert, to reach the open state. Despite missing delays in the onset of recovery, the model appears to be adequate for phase-plane considerations for many excitable systems."
^{[3]}

## Physiological description

The Morris–Lecar model is a two-dimensional system of nonlinear differential equations. It is considered a simplified model compared to the four-dimensional Hodgkin-Huxley model.

Qualitatively, this system of equations describes the complex relationship between membrane potential and the activation of ion channels within the membrane: the potential depends on the activity of the ion channels, and the activity of the ion channels depends on the voltage. As bifurcation parameters are altered, different classes of neuron behavior are exhibited. <math>\tau_N</math> is associated with the relative time scales of the firing dynamics, which varies broadly from cell to cell and exhibits significant temperature dependency.^{[3]}

Quantitatively:

<math> C \frac{dV}{dt} = I - g_L (V-V_L) - g_{Ca} M_{ss} (V-V_{Ca}) - g_K N (V-V_K) </math>

<math> \frac{dN}{dt} = \frac{N-N_{ss}}{\tau_{N}}</math>

where

<math> M_{ss} = (1 + \tanh[(V-V_1)/V_2])/2 </math>

<math> N_{ss} = (1 + \tanh[(V-V_3)/V_4])/2 </math>

<math> \tau_N = 1/[\phi \cosh[(V-V_3)/(2V_4)]</math>

### Variables

- <math> V\ </math> : membrane potential
- <math> N\ </math> : recovery variable: the probability that the K+ channel is conducting

### Parameters and constants

- <math> I\ </math>: applied current
- <math> C\ </math> : membrane capacitance
- <math> g_L,\ g_{Ca},\ g_K </math> : leak, Ca++, and K+ conductances through membranes channel
- <math> V_L,\ V_{Ca},\ V_K </math> : equilibrium potential of relevant ion channels
- <math> V_1,\ V_2,\ V_3,\ V_4 </math>: tuning parameters for steady state and time constant

### Notes

- The <math>M_{ss}</math> and <math>N_{ss}</math> equations may also be expressed as <math>M_{ss} = 1 + \exp[-2(V-V_1)/V_2]^{-1}</math> and <math>N_{ss} = 1 + \exp[-2(V-V_3)/V_4]^{-1}</math>, however most authors prefer the form using the hyperbolic functions.

## Bifurcations

Bifurcation in the Morris–Lecar model have been analyzed with the applied current <math> I\ </math>, as the main bifurcation parameter and <math> \phi,\ g_{Ca},\ V_3,</math> <math> V_4 </math> as secondary parameters for phase plane analysis.^{[4]}

## See also

- Computational neuroscience
- Biological neuron model
- Hodgkin-Huxley model
- FitzHugh-Nagumo model
- Neural oscillations
- Nonlinear dynamics
- Quantitative models of the action potential

## External links

## References

- ↑
^{1.0}^{1.1}Morris C, Lecar H (1981). "Voltage Oscillations in the barnacle giant muscle fiber". Biophys J. 35(1): 193–213. - ↑ Keynes RD, Rojas E, Taylor RE, and Vergara J. 1973. "Calcium and potassium systems of a giant barnacle muscle fibre under membrane potential control". J. Physiol. (Lond.) 229:409-455.
- ↑
^{3.0}^{3.1}Harold Lecar (2007), Scholarpedia, 2(10):1333. - ↑ Tsumoto K et al (2006). "Bifurcations in the Morris-Lecar Model". Neurocomputing 69, 4-6: 293-316.