# Conductance-based models

Post-publication activity

Curator: Frances K. Skinner

Conductance-based models are the simplest possible biophysical representation of an excitable cell, such as a neuron, in which its protein molecule ion channels are represented by conductances and its lipid bilayer by a capacitor.

## Theoretical Basis

Conductance-based models are based on an equivalent circuit representation of a cell membrane as first put forth by Hodgkin and Huxley (1952). These models represent a minimal biophysical interpretation for an excitable cell in which current flow across the membrane is due to charging of the membrane capacitance, $$I_C \ ,$$ and movement of ions across ion channels. In its simplest version, a conductance-based model represents a neuron by a single isopotential electrical compartment, neglects ion movements between subcellular compartments, and represents only ion movements between the inside and outside of the cell. Ion channels are selective for particular ionic species, such as sodium ($$Na$$) or potassium ($$K$$), giving rise to currents $$I_{Na}$$ or $$I_K \ ,$$ respectively. Thus, the total membrane current, $$I_m(t) \ ,$$ is the sum of the capacitive current and the ionic current,

$$I_m(t) = I_C + I_{ionic} \ ,$$ where

$$I_C = C_m dV(t)/dt \ .$$

In the Hodgkin-Huxley model, the original conductance-based model,

$$I_{ionic} = I_{Na} + I_K + I_L \ .$$

The leak current, $$I_L \ ,$$ approximates the passive properties of the cell. Each ionic current is associated with a conductance (inverse of resistance) and a driving force (battery) which is due to the different concentrations of ions in the intracellular and extracellular media of the cell. Thus,

$$I_{ionic} = g_{Na}(V) [V(t) - V_{Na}] + g_K(V) [V(t) - V_K] + g_L [V(t) - V_L] \ .$$

This is illustrated in Figure 1. The voltage dependence or non-constant nature of the conductance, $$g$$ (1/resistance) of ion channels is captured using "activation" and "inactivation" gating variables which are described using first-order kinetics. This is represented with an arrow across the resistor in the schematic representation of Figure 1. A current due to ionic species $$S$$ with an activation gating variable, $$a \ ,$$ but no inactivation variable, would be given by $$g_S = \overline{g}_S \times a \ ,$$ where $$a$$ is described by first-order kinetics and $$\overline{g}_S$$ represents the maximal conductance for the particular ion channel.

## Formulation, Parameters and Assumptions

From the theoretical basis described above, the standard formulation for a conductance-based model is given as$C_m dV/dt = \Sigma_j g_j (V_j - V) + I_{ext}$

where

$$g_j = \overline{g}_j a_j^x b_j^y$$ with

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

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

for each j. $$V_j$$ is the Nernst potential or reversal potential for current $$j \ ,$$ $$(V - V_j)$$ is called the driving force for $$j \ ,$$ and $$I_{ext}$$ is an external current that may be present. $$a, b$$ are gating variables raised to small integer powers $$x, y \ ,$$ respectively. $$a_{\infty}, b_{\infty}$$ are the steady-state gating variable functions that are typically sigmoidal in shape. $$\tau$$ 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 (ODEs), 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 $$V \ ,$$ 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 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 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:

1. the different ion channels in the cell membrane are independent from each other,
2. activation and inactivation gating variables are voltage-dependent and independent of each other for a given ion channel type,
3. each gating variable follows first-order kinetics, and
4. the model cell compartment is isopotential.

## Examples and Variants

• Hodgkin-Huxley model (1952): Original conductance-based model based on the giant axon of the squid producing action potentials. There is a sodium current with activation and inactivation variables, a potassium current with only an activation variable, and a (passive) leak current.
• Connor-Stevens model (1971): Extended action potential generating model using gastropod neuron somas. There is a sodium, potassium and leak current as in the Hodgkin-Huxley model, and in addition, another potassium current that is transient, the so-called A-current, is included. This current has an activation and an inactivation variable.
• Morris-Lecar model (1981): Based on the barnacle muscle fiber. There is a calcium current with an instantaneous activation, a potassium current with an activation variable, and a (passive) leak current.

Conductance-based models are the most common formulation used in neuronal models and can incorporate as many different ion channel types as are known for the particular cell being modeled. A common extension found in many conductance-based models is the inclusion of an equation for calcium dynamics. Ionic currents can be calcium-dependent in addition to voltage-dependent with calcium concentrations being controlled by calcium currents, pumps and exchangers. For example, see section 6.2 in Dayan and Abbott (2001).

Furthermore, as details of various ion channels are determined, variants of conductance-based models have been developed to better match the experimental data. For example, the standard conductance-based formalism derived from Hodgkin-Huxley models has been extended to account for state-dependent inactivation without voltage dependence in fast sodium and Kv3 potassium channels (Marom and Abbott 1994).

The simplest conductance-based model formulation from a spatial perspective consists of a single, isopotential compartment. Ion movement is strictly between the inside and outside of the cell. However, to incorporate spatial complexity of cells, several compartments can be connected to represent the cell's complex morphology (see compartmental model). A conductance-based model formulation can then be used for each compartment with additional terms added to the equations to represent the connections. That is, current flow occurs not only between the inside and outside of the cell, but also between different regions of the cell.

## Other Issues

Conductance-based models for excitable cells are developed to help understand underlying mechanisms that contribute to action potential generation, repetitive firing and bursting (i.e., oscillatory patterns) and so on. In turn, these intrinsic characteristics affect behaviors in neuronal networks.

However, as the number of currents included in conductance-based models expands, it becomes more difficult to understand and predict the resulting model dynamics due to the increasing number of differential equations. For example, the original Hodgkin-Huxley model is a 4th order system of ODEs. Efforts have been made not only to capture the qualitative dynamics of conductance-based models (e.g., FitzHugh-Nagumo model) but also to reduce the complexity of the system (e.g., Kepler et al. 1992).

Mathematical distinctions in conductance-based models using dynamical system and bifurcation analyses are available. Details are described in Izhikevich (2007).