A Theoretical Overview - Computational Neuroscience: A Comprehensive Approach - page 5

the calcium channel. In Chapter 3, the authors present a detailed review on
modelling calcium.

The model exhibits different behaviours in response to external current inputs.
A response may be described by an F − I curve, where F is the output fre-

quency and I is the input current. When F I is continuous, it is termed Type I

neuron (the integrate-and-fire model is of this type); when F

−

I is not continu-
ous, it is Type II neuron (the Hodgkin-Huxley is of this type). Mathematically,
the Type I neuron is usually due to a saddle-node bifurcation, whereas the Type
II neuron is due to a Hopf bifurcation.

−

Cable model In point models, we ignore the geometric properties of cells. One
way to include cell morphology in modelling is to treat cell segments as cylinders,
as described in the cable Equation (1.11).
Multi-compartment models Another way to include cell morphology is with a
multi-compartment model. According to actual neuronal anatomy, a biophysical
model of a few hundred compartments may be appropriate and it is a formidable
task to analyze such a model. Models with only a few compartments have been
investigated and it is surprising that such simplified models usually fit more compli-
cated models well [12, 56]. Here we consider two-compartment models, composed
of a somatic and a dendritic compartment.

• A two-compartment abstract model (see the integrate-and-fire model before).

When the somatic membrane potential V s (

t ) is below the threshold V thre ,

⎧
⎨

g (

g c V d (

t ) − V s ( t )

dV s ( t ) / dt = − V s ( t ) − V rest )+

p

(1.3)

g (

g c V s

( t ) − V d ( t ) I

⎩

dV d ( t ) / dt = − V d ( t ) − V rest )+ +

1 − p 1 − p

where 1 g is the decay rate, p is the ratio between the membrane area of the

somatic compartment and the whole cell membrane area, V d is the membrane
potential of the dendritic compartment, g c >

/

0 is a constant.

The properties of the two-compartment model above have been examined by a
few authors, see for example [62]. We have reported that a two-compartment
model is naturally a slope detector [21, 34, 46].

• A two-compartment biophysical model. A simplified, two-compartment bio-

physical model, proposed by Pinsky and Rinzel [56] is described here. They
have demonstrated that the model mimics a full, very detailed model of pyra-
midal cells quite well.