Information Technology Reference
In-Depth Information
To compute the dynamics of the membrane's PDE model without knowledge of
initial conditions and through the processing of noisy measurements, the following
stages are followed: Using the method for numerical solution of the PDE through
discretization the initial PDE is decomposed into a set of nonlinear ordinary
differential equations with respect to time [
139
]. Next, each one of the local
models associated with the ordinary differential equations is transformed into a
model of the linear canonical (Brunovsky) form through a change of coordinates
(diffeomorphism) which is based on differential flatness theory. This transformation
provides an extended model of the nonlinear system for which state estimation is
possible by application of the standard Kalman Filter recursion [
122
,
200
]. Unlike
other nonlinear estimation methods (e.g., Extended Kalman Filter) the application
of the standard Kalman Filter recursion to the linearized equivalent of the nonlinear
PDE does not need the computation of Jacobian matrices and partial derivatives and
does not introduce cumulative numerical errors [
157
-
160
,
170
].
6.2
Propagating Action Potentials
The Hodgkin-Huxley equation, in the cable PDE form, is re-examined. It holds that:
@
2
V
C
m
@
@t
D
4d
R
i
@x
2
I
ion
C I
(6.1)
@
@t
D ˛
.V/.1 / ˇ
.V/
where D m;h;, that is parameters that affect conductivity and consequently
the input currents of the membrane. If c is the velocity of propagation of the wave
solution, then one can use the notation
V.x;
ct
;t/D V.x/
(6.2)
By performing a change of coordinates D x ct one has
@
2
V
@
2
C
m
@
@t
D C
m
c
@
@
C
4d
R
i
C I I
ion
(6.3)
@
@t
D c
@
@
C ˛
.V/.1 / ˇ
.V/
where again D m;h;. The following boundary condition holds V. D˙1/ D
V
rest
. Equation (
6.3
) is decomposed into a set of equivalent ordinary differential
equations
dV
d
D U
dU
R
i
d
D
4d
.I
ion
I cC
m
U/
(6.4)
d
d
D.˛
Gamma
.V/.1 / ˇ
.V//=c
