View source for Conductance-based models

== Formulation, Parameters and Assumptions ==

From the theoretical basis described above, the standard formulation for a conductance-based model is given as:

<math> C_m dV/dt = \Sigma_j g_j (V_j - V) + I_{ext} </math>

where

<math> g_j = \overline{g}_j a_j^x b_j^y </math> with 

<math> da/dt = [a_{\infty}(V) - a]/\tau_a(V) </math> and

<math> db/dt = [b_{\infty}(V) - b]/\tau_b(V) </math> 

for each j. <math> V_j </math> is the [[Nernst potential]] or reversal potential for current <math> j \ ,</math> <math> (V - V_j) </math> is called the driving force for <math> j \ ,</math> and <math> I_{ext} </math> is an external current that may be present. <math> a, b </math> are gating variables raised to small integer powers <math> x, y \ ,</math> respectively. <math> a_{\infty}, b_{\infty} </math> are the steady-state gating variable functions that are typically sigmoidal in shape. <math> \tau </math> is the time constant, which can be voltage-dependent. Further details and equation descriptions can be found in many texts such as Hille (2001) and Koch (1999). Thus, conductance-based models consist of a set of ordinary differential equations ([[ODE]]s), as derived from current flow in a circuit representation following [[Kirchoff's laws]].  The number of differential equations in the set of model equations depends on the number of different ion channel types being represented with their particular activation and inactivation gating variables. The conductances can depend not only on transmembrane potential <math> V \ ,</math> but also on concentrations of different ions, for example, the concentration of calcium.
 
The parameters in conductance-based models are determined from empirical fits to voltage-clamp experimental data (e.g., see Willms 2002), assuming that the different currents can be adequately separated using pharmacological manipulations and [[Dynamic Clamp|voltage-clamp]] protocols.  As shown in the model formulation, the activation and inactivation variables can be raised to a non-unity integer power, and this is dictated by empirical fits to the data.  

Since (i) it is rarely possible to obtain estimates of all parameter values in a conductance-based mathematical model from experimental data alone, and (ii) the model construct is necessarily a simplification of the biological cell, it is important to consider various [[Neuronal Parameter Optimization|optimization techniques]] to help constrain the problem for which the conductance-based model was developed to address.

In summary, the basic assumptions in conductance-based models are: 
# the different ion channels in the cell membrane are independent from each other,  
# activation and inactivation gating variables are voltage-dependent and independent of each other for a given ion channel type, 
# each gating variable follows first-order kinetics, and
# the model cell compartment is isopotential.