Morris-Lecar Model

• Harold Lecar, University of California, Berkeley, California

The Morris-Lecar model is a two-dimensional "reduced" excitation model applicable to systems having two non-inactivating voltage-sensitive conductances. The original form of the model employed an instantaneously responding voltage-sensitive Ca²⁺ conductance for excitation and a delayed voltage-dependent K⁺ conductance for recovery. The equations of the model are: \[\tag{1} CV' = - g_{\rm Ca} M_{\rm ss} (V) (V -V_{\rm Ca}) - g_{\rm K}W (V -V_{\rm K}) - g_{\rm L}(V - V_{\rm L}) + I_{\rm app} \]

\[\tag{2} W' = (W_{\rm ss} (V)-W)/ T_{\rm W}(V) \ .\]

Here, \(V\) is the membrane potential, \(W\) is the "recovery variable", which is almost invariably the normalized K⁺-ion conductance, and \(I_{\rm app}\) is the applied current stimulus. The variable normalized conductance, \(W(t)\ ,\) is equal to the instantaneous value of the probability that a K⁺- ion channel is in its open (conducting) state. Eqn. (2) thus describes the relaxation process by which protein channels undergo conformational transitions between ion-conducting and non-conducting states. The key to electrical excitability is that the energies and transition rates for this channel-gating process are steeply voltage dependent.

This model was named after Cathy Morris and Harold Lecar, who derived it in 1981. Because it is two-dimensional, the Morris-Lecar model is one of the favorite conductance-based models in computational neuroscience.

Physically, the open-state probability functions, \(M_{\rm ss} (V)\) and \(W_{\rm ss} (V)\ ,\) are derived from the assumption that, in equilibrium, the open and closed states of a channel are partitioned according to a Boltzmann distribution. The energy difference between these states depends on the work required to translocate certain highly-charged membrane-spanning helices against the very strong trans-membrane electric field. Explicitly, the conductance functions are given as

\[\tag{3} M_{\rm ss} (V) = (1 + \tanh [(V - V_1)/V_2)])/2\ ,\]

\[\tag{4} W_{\rm ss} (V) = (1 + \tanh [(V - V_3)/V_4)])/2\ .\]

The function \((1+\tanh(x))/2\) in these equations could have been written more simply as \((1 + \exp(-2x))^{-1}\ ,\) but most of the literature preserves the form given above. The time constant for the K⁺- channel relaxation in response to changes of voltage is voltage-dependent,

\[\tag{5} T_W(V) = T_0 {\rm sech}[(V-V_3)/2V_4]\]

Here the parameter \(T_0\) sets the time scale for the recovery process. \(T_0\) can vary over a wide range for different cells, and is also extremely temperature sensitive. The relaxation process of Eqns. (4) and (5) is analogous to dielectric relaxation in an electric field. However, internal trans-membrane electric fields, can be so great that the energy difference between the open and closed states can be much greater than thermal energy (\(Q_{\rm gate}V\gg kT\)).

Eqns. (1) and (2) constitute a very simple model of excitability depending on three ionic currents\[I_{\rm Ca}\ ,\] which causes the initial excitation; \(I_{\rm K}\ ,\) the main current involved in recovery; and \(I_{\rm L}\) the membrane leakage current involved in maintaining the resting potential. Many different systems and excitability phenomena can be modeled by varying the magnitudes of these three currents (i.e., the peak conductances, \(g_{\rm Ca}\ ,\) \(g_{\rm K}\) and \(g_{\rm L}\)). These conductances can be modified in vivo by a number of means, such as varying specific ion concentrations or by pharmacological dissection with specific channel-blocking molecules. In this way both the experimental preparations and the theoretical models can be used to demonstrate a rich variety of qualitatively different behaviors.

Principal Assumptions

Generally, excitable systems have more than two relevant excitation variables, because there are often more than two species of gated channels and also because some channels have autonomous inactivation processes. Thus the primary assumption in using a two-dimensional model is that the true higher-order system can in fact be projected onto a two-dimensional phase space without altering the topological properties of the phase profile. This is true for the four-dimensional Hodgkin-Huxley system, which has a single singular point and exhibits excitation phenomena that can all be duplicated in two dimensions. There are other neural excitation phenomena such as bursting oscillations or chaotic firing which are intrinsically higher-dimensional, and cannot be duplicated in the phase plane.

The principal assumptions underlying the Morris-Lecar model include:

• 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.

Explained Phenomena

All of the parameters in the Morris-Lecar equations are experimentally measurable. Thus this simple model lends itself to simulating the rather wide range of phenomena that occur in different excitable systems.

With parameters appropriate to a nerve axon, the model yields prototypical single-shot firing, with a quasi-threshold, and an abrupt transition to repetitive firing over a narrow frequency range, as expected from the Hodgkin-Huxley or FitzHugh-Nagumo equations. Using parameters appropriate to a weak excitation current, the model displays spike trains emerging from zero frequency and oscillating over a relatively wide range of frequencies. These two types of behavior don't exhaust the repertoire of possible excitation. The system can also yield non-oscillatory bi-stable behavior with a true (saddle point) threshold, and various modes of pacemaker-like small oscillations.

Since the functions \(M_{\rm ss}(V)\) and \(W_{\rm ss}(V)\) are chosen to fit the actual voltage-sensitive conductances, the subsidiary parameters \(V_1,...,V_4\) can be fit to different systems of interest to alter the phase-plane properties of the system. Thus \(V_1\) and \(V_3\) set the steepness of the conductance curves, which, in turn, is determined by the effective charge moved during the gating transition. Real channels have effective gating charges varying from 5 to 14 electron charges, corresponding to \(V_2\) or \(V_4 =\) 5 to 2 mV. The parameters \(V_1\) and \(V_3\) locate the center (inflection point) of the conductance curves. The conductance curves can be translated along the voltage axis by varying the ionic strength or Ca²⁺ concentration of the electrolyte medium, thus altering the membrane surface- charge shielding of applied voltage.

The Morris-Lecar equations essentially describe "push-pull" relaxation oscillations, with \(g_{\rm Ca}\) and \(g_{\rm K}\) and the corresponding equilibrium potentials, \(V_{\rm Ca}\) and \(V_{\rm K}\) determining the relative strengths of "push" to pull. If the excitable conductance, \(g_{\rm Ca}\ ,\) is too great relative to the other conductances, the system may not be able to recover from a depolarization, and may exhibit bistability. \(I_{\rm app}\) is an applied current stimulus, which persistently depolarizes the cell. A critical value of \(I_{\rm app}\) can produce a transition from single-shot behavior to repetitive firing. The dynamics can also be altered dramatically by temperature changes, because the gating relaxation time \(T_0\) is exquisitely temperature sensitive (\(Q_{10} \approx 3\)), whereas the other time constants depend on peak ionic conductances which are not very temperature sensitive.

The striking feature, which motivated the original experiments that led to the model (Morris & Lecar, 1981), is the ability of the barnacle giant muscle fiber to undergo transitions to a plethora of oscillating states. By changing parameters, the Morris-Lecar equations could describe the two main classes of nerve oscillations first described by Hodgkin (1948) as well as other types of behavior.

Applications to Neural Modeling

The original Hodgkin-Huxley equations constituted an exact model of a motor neuron,the squid giant axon. The axon has two voltage-gated channels and a leakage current, but requires four-dimensions to describe the voltage and three conductance-relaxation processes. The Morris-Lecar approximation describes the same three currents by just two dynamical variables. In this approximation, the Na⁺ inactivation process is omitted, but can be subsumed by taking the recovery process to be artificially rapid. The other approximation is to make Na⁺ activation respond instantaneously. This generally doesn't cause problems because Na⁺ activation generally responds more rapidly than the membrane capacitative time constant, which sets the limit to how rapidly the membrane potential can change in response to an applied current.

The two-dimensional model can then readily be analyzed using phase-plane methods. In the phase plane, dynamical transitions correspond to geometric features of the nullclines. Such transitions as the one from single-shot firing to a stable limit cycle are seen as the parameter representing applied-current stimulus is varied to reach a point of bifurcation between the two types of dynamic solution. At the bifurcation, a linearized model of the equations will always have one real eigenvalue that goes through zero (saddle-node bifurcation).

The Morris-Lecar equations are particularly useful for modelling fast-spiking neurons, such as the pyramidal neurons of the neocortex. Pyramidal neurons exhibit true all-or-none firing, continuous trains in response to constant-current stimulation, onset of repetitive firing with arbitrarily low firing frequency, and a linear or square-root current-frequency relation. Such neurons are capable of coding of information via limit cycles, which may help explain the computational efficacy of neural systems which often must work with noise components of as high intensity as the signal strength. The Morris-Lecar equations with appropriate parameters have unbounded variance in inter-spike interval length, and so can be used as a generic model to show how both frequency and precise timing are employed for coding [Gutkin and Ermentrout, 1998]. An example is the modeling of coincidence detection in auditory neurons (Agnon-Snir, H., Carr, C.E. and Rinzel J., 1998).

In studies of neural integration and neural information processing, it may be necessary to study the behavior of model neurons in the presence of noise. For single-shot neurons with discontinuous threshold and unbounded latency, firing fluctuations in the presence of electrical noise can be described by a random walk in the phase plane, with sub-and supra-threshold trajectories separated in the neighborhood of a threshold separatrix (Lecar and Nossal, 1970). Finer details of behavior in the presence of noise, such as latency fluctuations, spike interval fluctuations, and frequency jitter require models of neural dynamics that include the recovery process. Problems of interest that can be attacked are the question of whether the different classes of neural oscillator respond differently to input noise and the response of coupled arrays to noise.

Morris-Lecar-type models may prove useful for studying scaling phenomena, such as showing how neural oscillators and oscillatory networks change as the cells grow during development. Such problems on real systems, such as the lobster somatogastric ganglion, involve more species of channels than the two dimensional models, but the Morris-Lecar equations with ion-channel densities made explicit might give insight into the developmental features that allow oscillation phenomena to maintain their frequency as an animal grows.

Finally, the Morris-Lecar model neuron has been applied to modeling networks of coupled neural oscillators. Here the simple but realistic parameterization allows one to describe collective oscillations which depend on the inter-neuron coupling. A model employing Morris-Lecar oscillators of different frequencies has been used to explain quite complex bursting phenomena of coupled neurons (Buono, PL. and Golubitsky, M., 2001).

History

The Morris-Lecar equations grew out of an experimental study of the excitability of the giant muscle fiber of the huge Pacific barnacle, balanus nubilis (named by Charles Darwin, who probably meant to call it balanus nobilis). Invertebrate muscles are so heavily innervated that they don't require propagated action potentials to spread excitation throughout the cell. Synaptic depolarizations lead to the opening of Ca²⁺- channels, which are located in deep clefts of the surface membrane. The clefts allow external Ca²⁺ ions to enter deep into cell interior to activate muscle contraction. The clefts produce a variable resistance in series with parts of the excitable membrane, so that attempts to voltage clamp the fiber using an axial electrode often produce unwanted voltage oscillations.

In the early experiments, the inability to control the voltage turned out to be a virtue, because it forced the experimenters to devise a current-clamp analysis instead of the unattainable voltage-clamp. The current-clamped muscle fiber exhibits an entire zoo of different types of action potentials, plateau action potentials, and relaxation oscillations. The exact behavior of the system is extremely sensitive to small changes in experimental parameters. In order to explain this great variety of response, we made a simplified model of a membrane with the two major observed conductances, both non-inactivating. This model was certainly an oversimplification of the actual conductance processes in the barnacle, but the interplay of two non-inactivating conductances leads to a system whose qualitative phase portrait changes with small changes in experimental parameters, such as the relative density of Ca²⁺ and K⁺ channels or the relative relaxation-times of the conducting systems.

The simple two-conductance model was capable of simulating the entire panoply of (two-dimensional) oscillation phenomena that had been observed experimentally (Morris and Lecar, 1981). Parameter maps in phase-space can then be drawn to identify and classify the parametric regions having different types of stability. This engineering approach turned out to be fruitful, and seemed to have caught the interest of mathematician friends who were also interested in neural oscillations (Rinzel and Ermentrout, 1989).

Morris and Lecar's eponymous two-dimensional equations appeared in the middle of a longish paper. In fact, the real barnacle fiber exhibited many phenomena that required a third dimension, such as bursting oscillations and variable-duration plateaus, which terminated abruptly. One 3-D variant of the model employed a non-instantaneous (dynamical) activation and a non-linear instantaneous Ca²⁺ current-voltage curve to give exact quantitative fits to plateau action potentials. Another variant included internal Ca²⁺ ion accumulation and diffusion, which explained the observed abrupt terminations (in msec) of plateau action potentials, which can persist for many seconds. Because of the simplicity of the 2-D model, and the joys of studying the phase plane, the Morris-Lecar equations seemed to have taken on a life of their own as a prototypical model for excitability in a wide variety of systems.

Mathematical Properties

The Morris-Lecar equations share a common structure with a large class of two-dimensional excitability models, such as the \(V, n\) reduced Hodgkin-Huxley equations (FitzHugh, 1969) and the FitzHugh-Nagumo model (FitzHugh, 1961). In all of these models, excitation comes about as a jump phenomenon originating in a negative-resistance current-voltage relation. In one dimension the voltage is bistable, when the recovery is turned on, the upper stable state vanishes and one gets a conventional action potential.

The character of a particular Morris-Lecar system can be gauged by studying its properties in the \(V\ ,\) \(W\) phase plane. From Eqns. (1) and (2) the vertical and horizontal nullclines are given by

\[\tag{6} W_{\rm vert}[V] := (I_{\rm app} - g_{\rm Ca}M_{\rm ss}[V] (V - V_{\rm Ca}) - g_{\rm L}(V - V_{\rm L}))/(g_{\rm K}(V - V_{\rm K}))\ ,\]

\[\tag{7} W_{\rm hor}[V] := (1 + \tanh [(V - V_3)/V_4)])/2\ .\]

Since neither of these nullclines is linear, the two curves can intersect in a variety of ways as the conductance parameters are varied. The intersection of the vertical and horizontal nullclines, of course, gives the singular points for each different parameter set. As an example, the vertical nullcline can be altered systematically by taking a progression of values for \(g_{\rm Ca}\ .\) For the high and low values of \(g_{\rm Ca}\ ,\) the nullclines intersect once. For a narrow intermediate range of \(g_{\rm Ca}\ ,\) the nullclines can intertwine and intersect at three points. Whenever there three singular points, one of them must be a saddle point (from consideration of Poincare indices or just the demand that two sets of phase trajectories can't intersect).

Since neither of the voltage-dependent conductances inactivates, the character of the predicted response depends entirely on on the relative strength of \(g_{\rm Ca}\) compared to the other two conductances, \(g_{\rm L}\) and \(g_{\rm K}\ .\) The character of each singular point is determined by the roots of the characteristic equation for the eigenvalues of the equations linearized in the neighborhood of the singular point. These roots in turn are determined by a discriminant involving the elements of the Jacobian matrix evaluated at that point. These, in turn, are determined by the slopes and angles of intersection of the two nullclines at the singular point. Because of the nonlinearities of both nullclines, there are a number of different geometric possibilities for the intersections, and hence a surprising number of singular-point patterns. Thus the singular points can be stable or unstable, nodes or foci as determined by changes in the roots of the characteristic equation that are sensitive to modest changes in the conductance parameters.

The interesting feature of the particular nullcline shapes in the \(V, W\) phase plane is that a number of different distinct features of the two major oscillation types first described by Hodgkin (Hodgkin, 1948) can be shown to be linked by phase-plane considerations. Hodgkin's class I neurons are are characterized by a sharp threshold for firing, long latencies near threshold, and oscillations whose frequency can be varied over a relatively broad range starting from arbitrarily low frequency. Sensory neurons capable of frequency coding fall into this class. Class II neurons, such as the squid giant axon (and its theoretical counterpart -- the Hodgkin-Huxley model) are characterized by a variable threshold (FitzHugh's " quasi threshold"), short latencies, and abrupt transition to oscillation at a characteristic minimum frequency. The transition to oscillatory behavior of class I neurons is characteristic of of a saddle-point singularity, whereas the onset of repetitive firing of class II comes from a Hopf bifurcation at a critical value of stimulating current.This distinction is illustrated for the Morris-Lecar equations in a number of recent textbooks (Ellner and Guckenheimer, 2006, Izhikevich 2007); see also Neuronal Excitability.

References

• Agnon-Snir, H., Carr, C.E., and Rinzel, J. (1998) The role of dendrites in auditory coincidence detectors. Nature 393: 268 - 272.

• Buono, PL. and Golubitsky, M. (2001) Models of central pattern generators for quadruped locomotion. J. math. Biol. 4: 291-326.

• Edelstein-Keshet, L. (1988) Mathematical Models in Biology, Random House, New York.

• Ellner, S. P. and Guckenheimer, J. C. (2006) Dynamic Models in Biology, Princeton University Press.

• FitzHugh, R. (1961) Impulses and physiological states in models of nerve membrane. Biophys. J. 1: 445-466.

• FitzHugh, R. (1969) in Mathematical models for excitation and propagation in nerve. In: Biological Engineering, Schwann, H. P. (ed.) McGraw Hill, New York.

• Gutkin, B. and Ermentrout, G.B. (1998) Dynamics of membrane excitability determine interspike interval variability: a link between spike generation mechanisms and cortical spike train statistics. Neural Computation 10: 1047-1065.

• Hodgkin, A.L. (1948) The local electric changes associated with repetitive action I a non-medullated axon. J. Physiol. (London) 107: 165-181.

• Izhikevich, E. M. (2007) Dynamical systems in Neuroscience, MIT Press, Cambridge

• Marsalek, P. and Kubena, A.(2001) Phase space analysis of the 2D nonlinear system. 4th International neural coding Workshop.

• Morris, C. and Lecar, H. (1981) Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35: 193 - 213.

• Rinzel, J. and Ermentrout, B. (1998) Analysis of neural excitability and oscillations. Pages 251-292 in C. Koch and I. Segev (eds.) Methods of Neuronal Modeling, MIT Press, Cambridge.

• Tsumoto, K., et al (2002) Bifurcations in Morris-Lecar neuron model. International Symposium on Nonlinear Theory and its Applications

Internal references

• Yuri A. Kuznetsov (2006) Andronov-Hopf bifurcation. Scholarpedia, 1(10):1858.
• John Guckenheimer (2007) Bifurcation. Scholarpedia, 2(6):1517.
• Eugene M. Izhikevich (2006) Bursting. Scholarpedia, 1(3):1300.
• Frances K. Skinner (2006) Conductance-based models. Scholarpedia, 1(11):1408.
• James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
• Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
• Eugene M. Izhikevich and Richard FitzHugh (2006) FitzHugh-Nagumo model. Scholarpedia, 1(9):1349.
• Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
• Yuri A. Kuznetsov (2006) Saddle-node bifurcation. Scholarpedia, 1(10):1859.
• Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
• John W. Moore (2007) Voltage clamp. Scholarpedia, 2(9):3060.

See Also

Conductance-Based Models, FitzHugh-Nagumo Model, Hodgkin-Huxley Model, Neuronal Excitability

Sponsored by: Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia

Sponsored by: Frances K. Skinner, Toronto Western Research Institute, University Health Network, and University of Toronto

Reviewed by: Anonymous

Accepted on: 2007-08-27 21:46:05 GMT