Information Technology Reference
In-Depth Information
The Hodgkin-Huxley model is exclusively of electric nature: (1) The action
potential is based on the electric cable theory in which the pulse is a consequence of
voltage and time-dependent changes of the conductance for sodium and potassium,
(2) The nerve pulse consists of a voltage signal that changes both in space and time
and this change is driven by the flow of ions, through channels formed by proteins,
(3) The model is consistent with the channel-blocking effects of several poisons,
such as tetrodotoxin, but does not provide an explication of anesthesia (4) Reversible
changes in heat and mechanical changes are not explicitly addressed, and according
to this model heat generation should be expected.
The soliton model is of both electric and mechanical nature: (1) The propagation
of solitons is a consequence of the nonlinearity of the elastic constants close to
the melting transition of the lipid's membrane, (2) The model does not consider an
explicit role of protein and ion channels, (3) The propagating pulse is associated
with changes in all variables of the membrane including temperature, lateral
pressure, thickness, and length (4) In accordance with experimental findings, the
propagating pulse in this model does not dissipate heat, (5) the model provides an
explanation of the mechanism of anesthesia.
6.5
Estimation of Nonlinear Wave Dynamics
It will be shown that the new nonlinear filtering method, the so-called Derivative-
free nonlinear Kalman Filter, can be used for estimating wave-type dynamics
in the neuron's membrane. This can be done through the processing of noisy
measurements and without knowledge of boundary conditions. Previous results on
the application of Kalman Filtering in the estimation of neuronal dynamics can be
found in [
77
,
134
,
208
]
The following nonlinear wave equation is considered
@
2
@t
2
D K
@
2
(6.24)
@x
2
C f./
Using the approximation for the partial derivative
@
2
@x
2
'D
i
C
1
2
i
C
i
1
x
2
(6.25)
and considering spatial measurements of variable along axis x at points x
0
C
ix; i D 1;2; ;N one has
@
2
i
@t
2
K
2K
1
D
x
2
iC1
x
2
i
C
x
2
i1
C f.
i
/
(6.26)
By considering the associated samples of given by
0
;
1
; ;
N
;
NC1
one has
