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A new technique is proposed in this paper for real-time monitoring of brain neural activity based on the balloon model. A continuous-discrete extended Kalman filter is used to estimate the nonlinear model states. The stability, controlla- bility and observability of the proposed model are described based on the simulation and measured clinical data analysis. By introducing the controllable and observable states of the hemodynamic signal we have developed a numerical tech- nique to validate and compare the impact of brain signal parameters affecting on BOLD signal variation. This model increases significantly the signal-to-noise-ratio (SNR) and the speed of brain signal processing. A linear-quadratic regulator (LQR) also has been introduced for optimal control of the model.

In the brain, real-time monitoring of hemodynamic states and preserving their stability provides a significant mechanism for fast and reliable brain monitoring especially in early detection of seizure and epilepsy or in brainmachine interaction studies. In order to measure the neural activity of the brain the electroencephalography (EEG) and magnetoencephalography (MEG) could be applied for electrophysiological aspects and the functional magnetic resonance imaging (fMRI) and functional near infrared spectroscopy (fNIRS) [

At the same time the first compelling model for heomodynamic signal transduction in fMRI was presented in [

The work presented in [

Despite the widespread use of functional neuroimaging techniques [6,7,11,12], the physiological changes in the brain that accompanying neural activation are still poorly understood [2-7]. Due to the nonlinear and/or unspecified effects of different parameters on BOLD signal variation, there is no specific criterion to validate and observe the impact of each parameter.

The highly dependency and correlation of neurons processing, metabolic and vascular responses are conceptually well known in time and state space [

We have introduced an efficient hemodynamic state stimulation technique at [

As a consequence, in this paper the proposed model is introduced in Section 2. Section 3 presents the simulation and experimental results following by analysis and discussions on the stability, controllability and observability of the proposed system. A linear-quadratic regulator (LQR) also has been introduced at the end.

We have introduced an extended balloon model as depicted in

here is the mean transit time through the compartment, (0 - 30 s) indicated the viscoelastic time constant (inflation) and (0 - 30 s) is the viscoelastic time constant (deflation).

Equation (1) thereby introduces a fundamental nonlinearity, sufficient to generate all transients of the BOLD response. The variation of the normalized [HbR] concentration (q), can be defined as:

The core of the model is the physical necessity to largely increase CBF, to achieve a small increase in oxygen delivery. An increase in cerebral blood flow is very closely linked to the underlying neuronal activity [_{in}) and the output is the BOLD signal (y). The BOLD signal is partitioned into an extra and intravascular component, weighted by their respective volumes. These signal components depend on the deoxy-hemoglobin content and render the signal a nonlinear function of v and q.

By extending the model to cover the dynamic coupling of synaptic activity and flow a complete model, relating experimentally induced changes in neuronal activity to BOLD signal, obtains. Here we have considered four different states include: v cerebral blood volume (CBV), q deoxyhaemoglobin content, s flow inducing signal, f, CBF. These equations are acquired from the magnetic properties of hemoglobin which is diamagnetic for oxyhemoglobin and paramagnetic for deoxy-hemoglobin. Using the electromagnetic equations around a cylinder and variation with oxygen saturation, the balloon model can be obtained. The neural activity signal u is the input of the model. The mathematical expression of hemodynamic balloon model is as follows:

And output which is BOLD signal is:

We can measure a new parameter which is CMRO2 normalized to baseline too:

In this model is baseline oxygen extraction fraction, is baseline blood volume, is weight for deoxy Hb change and is the weight for blood volume. is the mean transit time of the venous compartment, α is the stiffness component of the balloon model, is the signal decay time constant, is the autoregulatory time constant, and ɛ is the neuronal efficacy. Now, we can describe the state space equations as a nonlinear dynamic system. The state of the system is a vector:

Extended Kalman filter is a nonlinear version of Kalman filter using for nonlinear dynamic systems, applied to estimate the states of balloon model. The nonlinear stochastic dynamical system is described by following state space equation:

This model includes perturbation and measurement noise, because of weak signal of fMRI. In this state space equation, is the state which is dependent on time, is input stimulus, is the perturbation noise (a white noise) which has mean 0 and variance Q. is measurement noise which is a white noise with mean 0 and variance R. The and are independent Gaussian sequences having the following properties:

The prediction is established as:

The Jacobian matrix and are defined as follow:

The input signal u in the balloon model is the neural activity and it is created by a square stimulus signal. The relation between the neural activity and the stimulus signal can be stated by:

where is the stimulus step signal and is an inhibitory feedback signal and k is a gain factor. is a time constant. So, first the neural activity will be produced from and then use neural activity as an input to the balloon model.

The balloon model is implemented in Simulink and a reasonable neural activity input is produced. Then the output is plotted as a BOLD signal. A white noise is added to this signal in order to mimic a noisy BOLD signal. Using proposed extended Kalman filter, the output due to the noisy signal follows the measurements (

The proposed system is verified using measured clinical data also plotted in

Bifurcation analysis investigates the stability of the system under change of parameters. Thus, it is important to first investigate nonlinear stability of the system and then use MatCont for Bifurcation analysis.

For bifurcation analysis first the equilibrium point should be calculated as follow:

As we see the important parameters for the equilibrium point are, , , and we have investigated their effects in the bifurcation analysis. Now, we put this equilibrium point in the Jacobian matrix and then find the eigenvalues of that matrix to investigate the nonlinear stability of balloon model for different parameters.

The eigenvalues of the Jacobian matrix in this case are:

As we see all of the eigen-values are negative and this system for every choice of parameters is always stable. This is very interesting achievement regarding to the stability characteristics of the proposed system to describe the hemodynamic parameters. These eigenvalues depend on, , , , and independent of. For the bifurcation analysis, in this paper MatCont is used to analyze stability of the system with change of different effective parameters of the system.

Bifurcation analysis shows that the nonlinear stability of balloon model is always guarantied. Here the stability is also represented based on the linearization of the balloon model. This is a simpler model of stability when all of the eigenvalues of the system () are negative. In this situation, the system is linearizable and it is possible to linearize the state space equation and then use the definition of stability, observability, and controllability in the linear case. The linear model is in the form of:

where:

and is the equilibrium point.

The eigenvalues of A are:

All of these eigenvalues are negative and equal to the case of nonlinear stability analysis when u = 0. So, the system is stable. The Bode diagram and the Root-Locus of the open-loop and closed-loop systems are shown in

The linear and nonlinear controllability and observability has been investigated at this section. For linear controllability and observability based on the controllability matrix

and observability index

we can determine their determinant to investigate if the system is controllable and observable or not. The determinant of each matrix is as followed:

For nonlinear controllability and observability, the nonlinear balloon model is directly investigated. For nonlinear controllability we have:

(23)

For nonlinear observability we have:

where:

The observability and controllability of linear and nonlinear systems depend on all previously introduced parameters, so they have been calculated in equilibrium point for all of these parameters. The calculation results show that for these values the system is controllable and observable.

As the system is controllable we can design an LQR controller. So, a full state feedback with a proper gain vector k can effectively control the system in a neighborhood of the equilibrium point. We consider the output to be matched with the experimental results. The input, output and the states of the system are shown in Figures 6 and 7, when the LQR controller is applied.

A new model for real-time monitoring of brain neural activity is proposed in this paper based on the balloon model. The stability, controllability and observability of the proposed model are described based on the simulation and measured clinical data analysis. By introducing the controllable and observable states of the hemodynamic signal we have developed a numerical technique to validate and compare the impact of brain signal parameters affecting on BOLD signal variation. This model increases significantly the SNR and the speed of brain signal processing. Up to our knowledge this is the first work on evaluation of these control parameters and introducing their practical impacts on clinical application. Surprisingly we realized that the system is always stable independent from any variation in blood flow and HbR/HbO variation.

The observability and controllability characteristics are introduced as significant factors to be considered as an evaluation tool to verify the preference of different hemodynamic factors. The preferred factors then can be considered based on their specified priority for further diagnosis and monitoring in clinical applications. This model can also be efficiently applied in any monitoring and control platform include brain and for study of hemodynamic and brain imaging modalities such as pulse-oximetry and fNIRS.