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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 muscle fiber of the giant barnacle . Morris–Lecar neurons exhibit both class I and class II
neuron A neuron, neurone, or nerve cell is an membrane potential#Cell excitability, electrically excitable cell (biology), cell that communicates with other cells via specialized connections called synapses. The neuron is the main component of nervous ...
excitability.


History

Catherine Morris (b. 24 December 1949) is a Canadian biologist. She won a Commonwealth scholarship to study at
Cambridge University , mottoeng = Literal: From here, light and sacred draughts. Non literal: From this place, we gain enlightenment and precious knowledge. , established = , other_name = The Chancellor, Masters and Schola ...
, where she earned her PhD in 1977. She became a professor at the
University of Ottawa The University of Ottawa (french: Université d'Ottawa), often referred to as uOttawa or U of O, is a bilingual public research university in Ottawa, Ontario, Canada. The main campus is located on directly to the northeast of Downtown Ottaw ...
in the early 1980s. As of 2015, she is an
emeritus professor ''Emeritus'' (; female: ''emerita'') is an adjective used to designate a retired chair, professor, pastor, bishop, pope, director, president, prime minister, rabbi, emperor, or other person who has been "permitted to retain as an honorary title ...
at the University of Ottawa. Harold Lecar (18 October 1935 – 4 February 2014) was an American professor of biophysics and neurobiology at the
University of California Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant u ...
. He graduated with his PhD in physics from
Columbia University Columbia University (also known as Columbia, and officially as Columbia University in the City of New York) is a private research university in New York City. Established in 1754 as King's College on the grounds of Trinity Church in Manha ...
in 1963.


Experimental method

The Morris–Lecar experiments relied on the voltage clamp method established by Keynes ''et al.'' (1973).


The principal assumptions underlying the Morris–Lecar model

Among the principal assumptions are these: #Equations apply to a spatially iso-potential patch of membrane. There are two persistent (non-inactivating) voltage-gated currents with oppositively biased reversal potentials. The depolarizing current is carried by Na+ or Ca2+ ions (or both), depending on the system to be modeled, and the hyperpolarizing current is carried by K+. #Activation gates follow changes in membrane potential sufficiently rapidly that the activating conductance can instantaneously relax to its steady-state value at any 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.


Physiological description

The Morris–Lecar model is a two-dimensional system of
nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...
. It is considered a simplified model compared to the four-dimensional
Hodgkin–Huxley model The Hodgkin–Huxley model, or conductance-based model, is a mathematical model that describes how action potentials in neurons are initiated and propagated. It is a set of nonlinear differential equations that approximates the electrical charac ...
. 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. is associated with the relative time scales of the firing dynamics, which varies broadly from cell to cell and exhibits significant temperature dependency. Quantitatively: : \begin C \frac & ~=~ I - g_\mathrm (V-V_\mathrm) - g_\mathrm M_\mathrm (V-V_\mathrm) - g_\mathrm N (V-V_\mathrm) \\ pt \frac & ~=~ \frac \end where : \begin M_\mathrm & ~=~ \frac \cdot \left(1 + \tanh \left[\frac \right">V-V_3)_(2V_4).html" ;"title="varphi \cosh[(V-V_3)/(2V_4)">varphi \cosh[(V-V_3)/(2V_4)/math> : : : --> : \begin M_\mathrm & ~=~ \frac \cdot \left(1 + \tanh \left[\frac \rightright) \\ pt N_\mathrm & ~=~ \frac \cdot \left(1 + \tanh \left[\frac \right]\right) \\ pt \tau_N & ~=~ 1 / \left( \varphi \cosh \left[\frac \right] \right) \end Note that the and equations may also be expressed as and , however most authors prefer the form using the hyperbolic functions.


Variables

* : membrane potential * : recovery variable: the probability that the K+ channel is conducting


Parameters and constants

* : applied current * : membrane capacitance * , , : leak, Ca++, and K+ conductances through membranes channel * , , : equilibrium potential of relevant ion channels * , , , : tuning parameters for steady state and time constant * : reference frequency


Bifurcations

Bifurcation in the Morris–Lecar model have been analyzed with the applied current , as the main bifurcation parameter and , , , as secondary parameters for phase plane analysis.


See also

*
Computational neuroscience Computational neuroscience (also known as theoretical neuroscience or mathematical neuroscience) is a branch of  neuroscience which employs mathematical models, computer simulations, theoretical analysis and abstractions of the brain to ...
* Biological neuron model *
Hodgkin–Huxley model The Hodgkin–Huxley model, or conductance-based model, is a mathematical model that describes how action potentials in neurons are initiated and propagated. It is a set of nonlinear differential equations that approximates the electrical charac ...
*
FitzHugh–Nagumo model The FitzHugh–Nagumo model (FHN), named after Richard FitzHugh (1922–2007) who suggested the system in 1961 and J. Nagumo ''et al''. who created the equivalent circuit the following year, describes a prototype of an excitable system (e.g., a n ...
* Neural oscillations *
Nonlinear dynamics In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
*
Quantitative models of the action potential In neurophysiology, several mathematical models of the action potential have been developed, which fall into two basic types. The first type seeks to model the experimental data quantitatively, i.e., to reproduce the measurements of current and volt ...


References


External links


A Morris–Lecar simulatorScholarpedia: Morris–Lecar ModelCatherine Morris – Research Profile
{{DEFAULTSORT:Morris-Lecar model Computational neuroscience