pjtchc1 November 3, 2005
1
Bonus Project: Making Your Own Neuron Circuits
A square-burster circuit was constructed in the LectureNotes folder:
matlab/mfilechcircuits/SquareBursterCircuit/PiecewiseLinearIV.
It used piecewise IV -characteristics for two nonlinear resistors in parallel,
one models a sodium Na+ channel and the other models a potassium K+
channel. The IV -characteristics for the Na+ channel has the shape of a
letter ‘S’ while that for the K+ channel has the shape of a letter ‘N ’. Also
the resting potential for the Na+ channel ENa is positive and that for the
K+ channel −EK is negative. The sodium-potassium pump is modeled by
an inductor and the bilipid layer of the cell wall is modeled by a capacitor.
All components are in parallel with each other. See illustration below.
Neuron Cell Membrane
Circuit Model
Equivalent Circuit and its IV -Characteristics
IV−Characteristics
1.5
I
g
1
0.5
E
0
V
K
C
ENa
−0.5
−1
−1
−0.5
0
0.5
1
1.5
2
2.5
3
pjtchc1 November 3, 2005
2
The ion current due to the chloride, Cl− , channel is assumed voltage
independent, therefore treated as a current source and lumped together with
the current term Iin . The solid IV -characteristic curve is the equivalence
of the Na+ and K+ characteristics in parallel with resisting potential EN a
and −EK respectively as shown. Mfiles, K.m, Na.m, KNa.m, from the folder
show how the characteristics are constructed using linear characteristics and
nonlinear ramp characteristics, which are piecewise linear.
In this project, instead of piecewise linear ramp characteristics, use either
the arctangent arctan x function or the tangent hyperbolic function tanh x
as basic ingredients for the ramp characteristics. For example, the following
curve
1
V − E0
1
arctan(
)+
I = −I0
π
V0
2
with parameter I0 , E0 , V0 would produces a ramp-like function for which the
following holds
lim I = −I0 ,
V →∞
lim I = 0, I(E0 ) = −
V →−∞
I0
.
2
The left graph below shows the shape of an arctangent ramp function.
0.1
I = −I0[arctan((V−E0) / V0) / π + 0.5]
−0.1
2
VC
0
I0 = 1
E0 = 0.5
V = 0.2
−0.2
−0.3
1
0
−1
−0.4
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
Ig
0.5
0
−0.6
−0.5
−0.7
6.5
−0.8
6.4
IL
−0.5
−0.9
−1
−1
1
1
0
6.3
−0.5
0
0.5
1
1.5
2
6.2
t
The goal is to produce smooth characteristics with configurations similar
to the one shown in the previous page, and to generate spike-bursts similar
to the one shown on the right figure above.