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the diffeomorphism (relations about state variables transformation) provided by
differential flatness theory.
Comparing to other nonlinear filtering approaches that could have been used,
the proposed filtering approach exhibits significant advantages: (1) it is based on
an exact linearization of the local ODEs thus avoiding numerical errors due to
approximative linearization, (2) in terms of computation it is faster than other
nonlinear filtering approaches, (3) it exhibits the optimality properties of the
standard Kalman Filter recursion for estimation under measurement noise.
6.6.2.3
Methods for Real-Time Monitoring of Neuron's Membrane
Dynamics
An established method for recording neuron's activity at low spatial resolution is
based on the use of multi-electrode arrays. Monitoring of the neurons' activity at
low resolution can be also obtained with the use of positron emission tomography.
This technique involves usage of radioactive tracers which bind to receptors. The
tracers emit rays which are detected by a scanning device and reflect the dynamics
of the binding sites. It has been also shown that it is possible to obtain voltage
measurements at high resolution from dendritic trees using fMRI techniques and
optical imaging. fMRI is a special case of magnetic resonance imaging. It is based
on the different magnetization features between oxygenated and non-oxygenated
tissue. fMRI provides recording of the activity of the neuron's membrane with high
spatial resolution (of the order of tens of m) and high sampling rate.
Optical methods are the most suitable for measuring voltage variations in neuron
membranes at high resolution. This can be succeeded, for example, through the
monitoring of molecules called chromophores or voltage sensitive dyes that change
their color according to voltage levels in the membrane. More recently, the use of
quantum dots as voltage signaling nanoparticles has enabled to further improve
the spatial resolution of optical imaging techniques. The resolution of the optical
imaging methods is at the m-scale and enables to obtain a sufficient number of
measurements for filtering and state estimation purposes [
123
-
138
].
6.6.2.4
Practical Applications of the Proposed Filtering Method
The proposed filtering method for distributed parameter systems has a wide range of
applications: (1) Estimation of the dynamics of the neuron's membrane with the use
of a small number of measurements. Due to practical constraints the spatiotemporal
variations of the membrane's voltage have to be performed using only a limited
measurements set. Moreover, knowledge about initial and boundary conditions
of the system is in most cases unknown. To estimate the dynamics of this type
of distributed parameter system the use of the proposed filtering method that is
based on the Derivative-free nonlinear Kalman Filter is of great importance, (2)
Estimation of the parameters of the models that describe the dynamics of biological
