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neurons. Actually, one has to do with a dual identification problem, that is estimation
of the non-measurable elements of the state vector of the biological neuron
and estimation of the unknown model parameters. The proposed Derivative-free
nonlinear Kalman Filter can offer an efficient solution to this problem (3) Change
detection about the parameters of the model that describes the neuron's dynamics.
This is a fault diagnosis problem and can result in efficient method of medical
diagnosis of neurological diseases. The filter can be parameterized to reproduce
the dynamics of the healthy neuron. By comparing real measurements from the
neuron against measurements obtained from the filter and by performing statistical
processing for the associated differences (residuals) one can diagnose deviation of
the neuron's functioning from the normal ranges. This can be particularly useful for
diagnosis of neurological diseases related to neurons decay, at their early stages.
Moreover, it is possible to perform fault isolation, which means to find which is the
distorted parameter in the neuron's model that is responsible for deviation from the
healthy functioning. This can be particularly useful for developing more efficient
medical treatments of neurological diseases (4) More efficient modelling of brain
functioning and of the nervous system. By detecting changes in the parameters of
neurons' model one can find out how the response and adaptation of the neurons'
functioning is associated with external stimuli and medication (pharmaceutical
treatment).
6.7
Conclusions
This chapter has analyzed wave-type PDEs that appear in the transmission of neural
signals and has proposed filtering for estimating the dynamics of the neurons'
membrane. It has been shown that in specific neuron models the spatiotemporal
variations of the membrane's voltage follow PDEs of the wave-type while in other
models such variations are associated with the propagation of solitary waves through
the membrane. To compute the dynamics of the membrane's PDE model without
knowledge of initial conditions and through the processing of a small number of
measurements, a new filtering method, under the name Derivative-free nonlinear
Kalman Filtering, has been proposed.
The method is based into decomposition of the initial wave-type PDE that
describes the dynamics of the distributed parameter system, into a set of nonlinear
ordinary differential equations. Next, with the application of a change of coordinates
(diffeomorphism) that is based on differential flatness theory, the local nonlinear
differential equations are turned into linear ones. This enables to describe the wave
dynamics with a state-space equation that is in the linear canonical (Brunovsky)
form. For the linearized equivalent of the wave system dynamics it is possible to
perform state estimation with the use of the standard Kalman Filter recursion. The
efficiency of the derivative free nonlinear Kalman Filter has been confirmed through
numerical simulation experiments.
