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Models for excitable cells and networks

Most models for excitable membrane retain the general Hodgkin-Huxley (HH) format [20], and can be written in the form:



where V denotes membrane potential (say, deviation from a reference, or ``rest'' level), C is membrane capacity, and is the sum of V- and t-dependent currents through the various ionic channel types; is the applied current. The variables are used to describe the fraction of channels of a given type which are in various conducting states (e.g., open or closed or ...) at time t. The first-order kinetics for typically involve V-dependence in the time constant ; is a temperature-like, time scale factor which may depend on i. If the current, , for channel type j may be suitably modelled as ohmic, then it might be expressed as


where is the total conductance with all j-type channels open (product of single channel conductance with the total number of j-channels), is the fraction of j-channels that are open (It may depend on several of the variables ), and is the reversal potential (usually Nernstian) for this ion species. For some channel types the current-voltage relation may be more appropriately represented by the Goldman-Hodgkin-Katz expression, or by a barrier-kinetics scheme [18], and the gating kinetics might involve a multi-state Markov description. In the classical HH model [20] for squid giant axon, there are three variables , denoted as m, h, and n, to describe the fractions, and , of open -channels and -channels, respectively.

For some purposes, it is important that the current balance equation (1) contain terms to account for ionic pump currents. These currents, as well as some channel conductances, may depend upon time-varying second messengers or ionic concentrations , e.g. in diffusionally-restricted intracellular and/or extracellular volumes. For such considerations, additional variables and transport/kinetic balance equations would be included in the model, and these will carry along their own time scales. Indeed, some models that include the dynamics of intracellular free calcium handling have assumed time constants which are orders of magnitude longer than channel kinetics and thereby set the time scale for phenomena like bursting oscillations (e.g., as in [2]). We also note that the form of (2) is not unique; in a phenomenological model of Rall (see [14]), the corresponding equations are nonlinear in the .

Some models for excitability contain many variables and represent numerous channel types, especially if one seeks to account for rather detailed aspects of spike shape and dependence upon many different pharmacological agents. On the other hand, if qualitative or semi-quantitative characteristics of spike generation and input-output relations are adequate, say in network simulations, then a reduced model having just a few variables may suffice. Such reductions can sometimes be obtained when time scale differences allow certain approximations such as relatively fast variables being instantaneously relaxed to pseudo-steady-state values, e.g. if is small relative to other time constants, then one might set in (2). Likewise, functionally related variables with similar time scales might be lumped together. In this spirit, FitzHugh [10] considered reductions of the HH model (also see [29] and [21]) and then introduced [11] an idealized, analytically tractable, two-variable model (also see [24]) which is widely studied as a qualitative prototype for excitable systems in many biological/chemical contexts. A FitzHugh-Nagumo/Hodgkin-Huxley hybrid was formulated and studied by Morris and Lecar [23], in the context of electrical activity of the barnacle muscle fiber. The model incorporates a V-gated -channel and a V-gated, delayed-rectifier, -channel; neither current inactivates. A simple version of this model is represented by the equations:





In equations (4)-(6), w is the fraction of -channels open, and the -channels respond to V so rapidly that we assume instantaneous activation. One might introduce dimensionless variables, as in [31] and [12], in order to (i) reduce the number of free parameters and identify equivalent groups of parameters, and (ii) identify and group ``fast" and ``slow" processes together. However, in the interest of clarity, we will keep all equations in their original form. In (5), has been scaled so its maximum is now one, and equals the temperature factor divided by the pre-scaled maximum ( in [23]). The V-dependent functions, , and , and the reference parameter sets are given in Appendix A. All the computations and figures in this chapter are based on equations (4)-(6), and extensions of them for generating bursting behaviors.

Even network models in certain approximations can reduce to a few variables. One example is the Wilson-Cowan model [37]: (for another, see Shamma in this volume)

Here, and represent the respective firing rates of a population of excitatory and inhibitory interneurons. The parameters are the membrane time constants; are the firing thresholds; are the ``synaptic weights''; and is a nonlinear saturating function similar in form to .

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Next: Understanding Dynamics via Up: No Title Previous: Introduction

Bard Ermentrout
Mon Jul 29 17:47:46 EDT 1996