Home | About | Journals | Submit | Contact Us | Français |

**|**Front Comput Neurosci**|**v.4; 2010**|**PMC2958054

Formats

Article sections

- Abstract
- Introduction
- The Mathematical Model
- Materials and Methods
- Results
- Discussion
- Conflict of Interest Statement
- References

Authors

Related links

Front Comput Neurosci. 2010; 4: 138.

PMCID: PMC2958054

Edited by: Klaus R. Pawelzik, University of Bremen, Germany

Reviewed by: H. T. Banks, North Carolina State University, USA; Franz Kappel, University of Graz, Austria

*Correspondence: Marìa Inés Troparevsky, Departamento de Matemática, Facultad de Ingenierìa, Universidad de Buenos Aires, Av. Paseo Colón 850,C1063ACV Buenos Aires, Argentina. e-mail: ra.abu.if@aportim

Received 2010 February 4; Accepted 2010 August 30.

Copyright © 2010 Troparevsky, Rubio and Saintier.

This is an open-access article subject to an exclusive license agreement between the authors and the Frontiers Research Foundation, which permits unrestricted use, distribution, and reproduction in any medium, provided the original authors and source are credited.

Sensitivity analysis can provide useful information when one is interested in identifying the parameter θ of a system since it measures the variations of the output *u* when θ changes. In the literature two different sensitivity functions are frequently used: the traditional sensitivity functions (TSF) and the generalized sensitivity functions (GSF). They can help to determine the time instants where the output of a dynamical system has more information about the value of its parameters in order to carry on an estimation process. Both functions were considered by some authors who compared their results for different dynamical systems (see Banks and Bihari, 2001; Kappel and Batzel, 2006; Banks et al., 2008). In this work we apply the TSF and the GSF to analyze the sensitivity of the 3D Poisson-type equation with interfaces of the forward problem of electroencephalography. In a simple model where we consider the head as a volume consisting of nested homogeneous sets, we establish the differential equations that correspond to TSF with respect to the value of the conductivity of the different tissues and deduce the corresponding integral equations. Afterward we compute the GSF for the same model. We perform some numerical experiments for both types of sensitivity functions and compare the results.

The electric process underlying the generation of the electroencephalography (EEG) signals can be modeled as a set of current sources within the brain. Considering the velocity of propagation of the electric waves in the brain, the static approximation of Maxwell's equations can be used to describe this process (see Hamalainen et al., 1993). The resulting model is a 3D Poisson-type equation with interfaces that relates the electric potential *u* in the head *G* with the impressed current *J _{i}* often represented by an electric dipole, i.e.,

In the literature two different sensitivity functions are frequently used: the traditional sensitivity functions (TSF) defined as u/θ that measures the variations of the output *u* when the parameter θ changes and the generalized sensitivity functions (GSF) that determine at which time instants the output of a dynamical system has more information about the value of its parameters in order to carry on an estimation process.

In this work we consider the head as a multi compartment nested set and assume homogeneity and isotropy, hence the conductivity function σ(*x*) that contains the value of the conductivity of the different tissues is approximated by a positive piecewise constant function. We analyze the sensitivity of the solution *u* to the FP with respect to the conductivity σ(*x*) by means of the TSF and the GSF and compare the results.

The paper is organized as follows. In Section “The Mathematical Model” we present the mathematical model. The TSF are introduced in Section “Materials and Methods.” Differential and integral equations are stated. In Section “Results,” we present the GSF. Based on Rubio et al. (2009) we calculate the GSF of *u* with respect to σ(*x*). We performed numerical experiments in a simple head model and present some conclusions.

The electrical activity of the brain consists of currents generated by biochemical sources at cellular level. The electric and magnetic fields that they produce can be estimated by means of Maxwell's equations (see Sarvas, 1987; Hamalainen et al., 1993). Based on the properties of the tissues involved, the velocity of propagation of the electromagnetic waves caused by potential changes within the brain is such that the effect may be detected simultaneously at any point in the brain or in the surrounding tissues. This fact justifies the use of a static approximation of Maxwell's equations. This approximation uncouples the equations for the magnetic and electric fields leading to the following second order partial differential equation (PDE)

$$\nabla \xb7(\sigma (x)\nabla u(x))=\nabla \xb7{J}_{i}(x)\text{\hspace{0.17em}\hspace{1em}}x\in G,$$

(1)

that relates the measured electric potential *u* and the impressed current *J _{i}*.

The volume *G* (head model) consists of nested homogeneous sets, one surrounded by the next one where the conductivity values are given.

We consider $G={\cup}_{j=1}^{3}{G}_{j}$, denoted from the inner one to the outer one by: *G*_{1}, the brain, *G*_{2}, the skull, and *G*_{3}, the scalp. We denote Γ =*G* the external surface of *G* and by *S*_{1} =*G*_{1}, *S*_{2} =*G*_{2} −*G*_{1} the surfaces between *G*_{1} and *G*_{2} and *G*_{2} and *G*_{3} respectively.

The positive, discontinuous and piecewise constant function σ(*x*) that contains the conductivity values of the different tissues at each point is:

$$\sigma (x)=\{\begin{array}{ccc}{\sigma}_{1}& x& \in {G}_{1}\\ {\sigma}_{2}& x& \in {G}_{2}\\ {\sigma}_{3}& x& \in {G}_{3}\\ 0& x& \notin G\end{array}.$$

(2)

Assuming that the potential and its normal derivative times the conductivity function are continuous across the transition surfaces, the resulting boundary value problem is:

$$\nabla \xb7(\sigma (x)\nabla u(x))=F(x)\text{\hspace{1em}}x\in G$$

(3)

$$\frac{\partial u(x)}{\partial \nu}=0,\text{\hspace{1em}}x\in \partial G$$

(4)

subject to:

$$[u]{{|}_{{s}_{i}^{-}}=0\text{\hspace{1em}}\left[\sigma (x)\frac{\partial u}{\partial \nu}\right]|}_{{S}_{i}^{-}}=0\text{\hspace{1em}}i=1,2.$$

(5)

where [·] denotes the difference between the values of the functions inside the brackets through the indicated surface, ν is the external normal vector and *F*(*x*) =·*J _{i}*(

If *F* is regular enough, there exist solutions (in a weak sense) to (3) provided that *F* verifies ${\int}_{G}^{}F(x)dx=0}.$ They are unique up to a constant (see Troparevsky and Rubio, 2003).

When *J*_{i} is an electric dipole, existence and uniqueness of solutions were studied in El Badia (2000).

The TSF is a common tool used to measure the variation of the output *u* of a system with respect to changes in its parameter θ =(θ_{1}, θ_{2},…,θ* _{p}*). Assuming smoothness of

$${s}_{i}(x)=\frac{\partial u}{\partial {\theta}_{i}}(x),\text{\hspace{1em}}i=1,\mathrm{\dots},p$$

(6)

The sensitivity functions are related to *u* by the Taylor approximation of first order. In the case of the system modeled by equations (1)–(5), considering *u* as a function of *x**G* and the vector of parameters θ =(σ_{1}, σ_{3,} σ_{3}), i.e., *u*=*u*(*x*, σ_{1}, σ_{1}, σ_{3,} σ_{3}) and fixing *x*, σ_{2}, σ_{3}, we have:

*u*(*x*, σ_{1} +δ, σ_{2,} σ_{3})*u*(*x*, σ_{1}, σ_{2}, σ_{3}) +*s*_{1}(*x*, σ_{1}, σ_{2}, σ_{3})δ

These functions *s _{i}*(

In the next subsection we derive the differential sensitivity equations (see Stanley, 1999; Burns et al., 2005) for the interface problem (3)–(5) introduced in Section “The Mathematical Model.”

We consider three parameters in our model: the conductivity values for each sub domain *G _{i}*, i.e., σ

$${s}_{1}(x)=\frac{\partial u}{\partial {\sigma}_{1}}(x),\text{\hspace{1em}}{s}_{2}(x)=\frac{\partial u}{\partial {\sigma}_{2}}(x),\text{\hspace{1em}}{s}_{3}(x)=\frac{\partial u}{\partial {\sigma}_{3}}(x)$$

These functions satisfy the following state equations that can be derived from (3) by formal differentiation (see Stanley, 1999):

$$\begin{array}{c}\begin{array}{cc}\Delta u+{\sigma}_{1}\Delta {s}_{1}=0& x\in {G}_{1}\\ \text{\hspace{1em}\hspace{1em}}{\sigma}_{2}\Delta {s}_{1}=0& x\in {G}_{2}\\ \text{\hspace{1em}\hspace{1em}}{\sigma}_{3}\Delta {s}_{1}=0& x\in {G}_{3}\end{array}\text{\hspace{0.05em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\begin{array}{cc}\text{\hspace{1em}\hspace{1em}}{\sigma}_{1}\Delta {s}_{2}=0& x\in {G}_{1}\\ \Delta u+{\sigma}_{2}\Delta {s}_{2}=0& x\in {G}_{2}\\ \text{\hspace{1em}\hspace{1em}}{\sigma}_{3}\Delta {s}_{3}=0& x\in {G}_{3}\end{array}\text{\hspace{1em}\hspace{1em}}\\ \begin{array}{cc}\text{\hspace{1em}\hspace{1em}}{\sigma}_{1}\Delta {s}_{3}=0& x\in {G}_{1}\\ \text{\hspace{1em}\hspace{1em}}{\sigma}_{2}\Delta {s}_{3}=0& x\in {G}_{2}\\ \Delta u+{\sigma}_{3}\Delta {s}_{3}=0& x\in {G}_{3}\end{array}\end{array}$$

The boundary condition for each *s _{i}* can be derived from (4) observing that the normal ν on the external surface Γ =

$$\frac{\partial {s}_{i}}{\partial \nu}{|}_{\partial G}=0\text{\hspace{1em}}i=1,2,3.$$

The equations (5) of continuity of *u* at the transition surfaces lead to:

$${s}_{i}{|}_{{S}_{j}^{-}}-\text{\hspace{0.05em}}{s}_{i}{|}_{{S}_{j}^{+}}=0,\text{\hspace{1em}}i=1,2,3.\text{\hspace{1em}}j=1,2.$$

and,

$$\begin{array}{l}\{\begin{array}{l}{\sigma}_{1}\frac{\partial {s}_{1}}{\partial v}{|}_{{S}_{1}^{-}}-\text{\hspace{0.17em}}{\sigma}_{2}\frac{\partial {s}_{1}}{\partial v}{|}_{{S}_{1}^{+}}=-\frac{\partial u}{\partial v}{|}_{{S}_{1}^{-}}\\ {\sigma}_{2}\frac{\partial {s}_{1}}{\partial v}{|}_{{S}_{2}^{-}}-\text{\hspace{0.17em}}{\sigma}_{3}\frac{\partial {s}_{1}}{\partial v}{|}_{{S}_{2}^{+}}=0\end{array}\\ \{\begin{array}{l}{\sigma}_{1}\frac{\partial {s}_{2}}{\partial v}{|}_{{S}_{1}^{-}}-\text{\hspace{0.17em}}{\sigma}_{2}\frac{\partial {s}_{2}}{\partial v}{|}_{{S}_{1}^{+}}=-\frac{\partial u}{\partial v}{|}_{{S}_{1}^{+}}\\ {\sigma}_{2}\frac{\partial {s}_{2}}{\partial v}{|}_{{S}_{2}^{-}}-\text{\hspace{0.17em}}{\sigma}_{3}\frac{\partial {s}_{2}}{\partial v}{|}_{{S}_{2}^{+}}=-\frac{\partial u}{\partial v}{|}_{{S}_{2}^{-}}\end{array}\\ \{\begin{array}{l}{\sigma}_{1}\frac{\partial {s}_{3}}{\partial v}{|}_{{S}_{1}^{-}}-\text{\hspace{0.17em}}{\sigma}_{2}\frac{\partial {s}_{3}}{\partial v}{|}_{{S}_{1}^{+}}=0\\ {\sigma}_{2}\frac{\partial {s}_{3}}{\partial v}{|}_{{S}_{2}^{-}}-\text{\hspace{0.17em}}{\sigma}_{3}\frac{\partial {s}_{3}}{\partial v}{|}_{{S}_{2}^{+}}=+\frac{\partial u}{\partial v}{|}_{{S}_{2}^{+}}\end{array}\end{array}$$

Note that in order to solve the PDE systems that correspond to the sensitivity functions *s _{i}*(

As for the equation (3), the sensitivity equations may only have solution in a weak sense.

Integral equations for the sensitivity functions can also be stated. They are useful to perform numerical approximations. We present the ones corresponding to *s*_{1}(*x*).

Recall that *s*_{1} satisfies:

$$\Delta u+{\sigma}_{1}\Delta {s}_{1}=0\text{\hspace{1em}}x\in {G}_{1}$$

(7)

$$\begin{array}{l}{\sigma}_{2}\Delta {s}_{1}=0\text{\hspace{1em}}x\in {G}_{2}\\ {\sigma}_{3}\Delta {s}_{1}=0\text{\hspace{1em}}x\in {G}_{3}\end{array}$$

(8)

with boundary condition:

$$\frac{\partial {s}_{i}}{\partial \nu}{|}_{\partial G}=0\text{\hspace{1em}}i=1,2,3.$$

(9)

and interface conditions:

$$\begin{array}{l}{s}_{1}{|}_{{S}_{j}}={s}_{1}{|}_{{S}_{j}^{+}},\text{\hspace{1em}}j=1,2,\text{\hspace{1em}}{\sigma}_{1}\frac{\partial {s}_{1}}{\partial \nu}{|}_{{S}_{1}^{-}}-\text{\hspace{0.17em}}{\sigma}_{2}\frac{\partial {s}_{1}}{\partial \nu}{|}_{{S}_{1}^{+}}=-\frac{\partial u}{\partial v}{|}_{{S}_{1}^{-}},\text{\hspace{1em}}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\sigma}_{2}\frac{\partial {s}_{1}}{\partial \nu}{|}_{{S}_{2}^{-}}-\text{\hspace{0.17em}}{\sigma}_{3}\frac{\partial {s}_{1}}{\partial \nu}{|}_{{S}_{2}^{+}}=0\end{array}$$

(10)

Applying green integral identity:

$$\underset{{G}_{i}}{\int}{s}_{1}\Delta \Phi -\Phi \Delta {s}_{1}dx=}{\displaystyle \underset{\partial {G}_{i}}{\int}{s}_{1}\nabla \Phi -\Phi \nabla {s}_{1}dS$$

(11)

in each *G _{i}*,

$$\begin{array}{l}{\displaystyle \sum _{i=1}^{3}\left[{\sigma}_{i}{s}_{1}(x)(-4\pi ){\chi}_{{G}_{i}}(x)-\sigma {\displaystyle \underset{{G}_{i}}{\int}\Phi \Delta {s}_{1}}dy\right]}\\ =\sigma {\displaystyle \underset{\partial {G}_{i}}{\int}{s}_{1}}(y)\frac{\partial \Phi}{\partial \nu}(x,y)dSy-{\sigma}_{i}{\displaystyle \underset{\partial {G}_{i}^{-}}{\int}\Phi (x,y)\frac{\partial {s}_{1}}{\partial \nu}(y)dSy.}\end{array}$$

(12)

Combining the former equation with (7), the boundary condition (9) and the continuity conditions at the transition surfaces (10) we obtain:

$$\begin{array}{l}-4\pi {s}_{1}(x)\chi {G}_{1}(x){\sigma}_{1}+{\displaystyle \underset{{G}_{1}}{\int}\Phi (x,y)\Delta}u(y)dy-4\pi {s}_{1}(x)\chi {G}_{2}(x){\sigma}_{2}\\ \text{\hspace{1em}}-4\pi {s}_{1}(x)\chi {G}_{3}(x)=({\sigma}_{1}-{\sigma}_{2}){\displaystyle \underset{{S}_{1}^{+}}{\int}{s}_{1}}\frac{\partial \Phi}{\partial v}dSy\\ \text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}}+{\displaystyle \underset{{S}_{1}^{+}}{\int}\frac{\partial u}{\partial v}\Phi dSy}+({\sigma}_{2}-{\sigma}_{3}){\displaystyle \underset{{S}_{2}^{+}}{\int}\frac{\partial \Phi}{\partial v}{s}_{1}}dSy\\ \text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}}+({\sigma}_{3}-{\sigma}_{4}){\displaystyle \underset{\partial {G}^{+}}{\int}{s}_{1}}(y)\frac{\partial \Phi}{\partial v}dSy.\\ \end{array}$$

(13)

If we take *x**G*_{1} in (13), since σ_{1}Δ*u*=*F* in *G*_{1} we obtain:

$$\begin{array}{l}{\sigma}_{1}{s}_{1}(x)=-\frac{1}{4\pi {\sigma}_{1}}{\displaystyle \underset{{G}_{1}}{\int}F(y)\frac{1}{|x-y|}}\\ -\frac{1}{4\pi}{\displaystyle \underset{{S}_{1}^{+}}{\int}\frac{\partial u}{\partial \nu}\Phi +\frac{{\sigma}_{2}-{\sigma}_{1}}{4\pi}}{\displaystyle \underset{{S}_{1}^{+}}{\int}{s}_{1}\frac{\partial \Phi}{\partial \nu}}\\ +\frac{{\sigma}_{3}-{\sigma}_{2}}{4\pi}{\displaystyle \underset{{S}_{2}^{+}}{\int}{s}_{1}\frac{\partial \Phi}{\partial \nu}\Phi dSy+}\frac{{\sigma}_{4}-{\sigma}_{3}}{4\pi}{\displaystyle \underset{{\Gamma}^{+}}{\int}{s}_{1}(y)\frac{\partial \Phi}{\partial \nu}dSy}.\end{array}$$

(14)

Similarly we can choose *x**G*_{2} or *x**G*_{3} to obtain the corresponding integral equations.

Remark Note that only the values of *s*_{1} on the transition surfaces appear in the right hand side of (14).

Letting $x\to \overline{x}\in {S}_{1}$ in (14), and recalling that:

$$\begin{array}{l}\underset{x\in \Omega \to \overline{x}\in \partial \Omega}{\mathrm{lim}}{\displaystyle \underset{\partial {\Omega}^{-}}{\int}f(y)\frac{x-y}{|x-y{|}^{3}}dSy}=-2\pi f(\overline{x})\\ +{\displaystyle \underset{\partial \Omega -\{\overline{x\}}}{\int}f(\overline{x})\frac{x-y}{|x-y{|}^{3}}dSy,}\end{array}$$

(15)

we can obtain an equation involving only values of *s*_{1} on the transition surfaces.

The GSF was introduced by Thomaseth and Cobelli (1999) to analyze qualitatively where the information about model parameters is concentrated during identification experiments. It was meant to understand how the estimation is related to observed system outputs, taking into account correlations between the model parameters. We recall the definition and some properties of the GSF, for details we refer to Thomaseth and Cobelli (1999).

Consider a non-linear parametric dynamical system:

$$\dot{u}(t)=f(t,u,\theta )$$

(16)

$$u({t}_{0})={u}_{0}$$

(17)

with a set of discrete time observations at *t*_{1}…*t*_{n}:

$${y}_{j}(\theta )=h({t}_{j},\theta )+{e}_{j},\text{\hspace{1em}}j=1,\mathrm{\dots},n$$

(18)

where *u*(*t*), *f*(*t*, *u*, θ),*R ^{N}*, θ

Let us consider the minimization of the weighted residual sum of squares:

$$J(y,{\theta}_{0})={\displaystyle \sum _{j=1}^{n}\frac{{\left({y}_{j}-h({t}_{j},{\theta}_{0})\right)}^{2}}{{v}_{j}^{2}}}.$$

(19)

A value $\widehat{\theta}$ where the minimum is attained is an estimator of θ_{0}.

The GSF at *t _{k}* are defined as:

$$\begin{array}{c}gs({t}_{k})={\displaystyle \sum _{j=1}^{k}\frac{1}{{\sigma}^{2}({t}_{j})}}[{F}^{-1}\times {\nabla}_{\theta}h({t}_{j},{\theta}_{0})]\xb7{\nabla}_{\theta}h({t}_{j},{\theta}_{0}),\text{\hspace{1em}}\\ k=1,\mathrm{\dots},n,\text{\hspace{1em}}l=1,\mathrm{\dots},p\end{array}$$

(20)

where *F* is the corresponding *p*×*p* Fisher information matrix:

$$F={\displaystyle \sum _{j=1}^{n}\frac{1}{{\sigma}^{2}({t}_{j})}{\nabla}_{\theta}h({t}_{j},{\theta}_{0}){\nabla}_{\theta}h{({t}_{j},{\theta}_{0})}^{T}}$$

(21)

and the symbol “·” in (20) represents element-by-element vector multiplication and Δ_{θ}*h* is the gradient that contains the derivatives of *h* with respect to each component of the parameter vector θ, that is the sensitivity of *h* with respect to the parameters. Thus, GSF are defined only at the discrete time points *t*_{1},…,*t*_{n} where measurements are taken. Note that *gs*(*t _{k}*) involves all the contributions of those measurements up to and including

$${\text{GSF}}_{\text{inc}}({t}_{k})={F}^{-1}\times \frac{{\nabla}_{\theta}h({t}_{k},\theta )}{{\sigma}^{2}({t}_{k})}\xb7{\nabla}_{\theta}h({t}_{k},\theta ),\text{\hspace{1em}}k=1,\mathrm{\dots},n$$

(22)

gives the information provided by the single measurements *t _{k}*. Note from (20) and (21) that the TSF have to be calculated in order to compute the GSF.

**Remark** The relationship between the GSF and the estimator of θ becomes clear if we introduce a continuous version of the parameter estimation problem supposing we have the measurements of the output for all *t*[0,*T*]. In this case the minimization of *J* in (19) can be thought in a space of probability (continuous version) leading to:

$$J(P,\theta )={\displaystyle \underset{0}{\overset{T}{\int}}\frac{y(t)-h{(t,\theta )}^{2}}{{v}^{2}(t)}}dP(t)$$

where *t*_{j}[0,*T*] and *P* is a probability function defined in [0,*T*]. In this frame the Fisher information matrix becomes ${F}_{ij}(T,\theta )={\int}_{0}^{T}{\scriptscriptstyle \frac{1}{{v}^{2}(t)}}{\scriptscriptstyle \frac{\partial h(t,\theta )}{\partial {\theta}_{i}}}{\scriptscriptstyle \frac{\partial h(t,\theta )}{\partial {\theta}_{j}}}dP(t)\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}$ and its inverse is, asymptotically and approximately, the covariant matrix of the estimator $\widehat{\theta}$ of θ. The generalized sensitivity at *t*[0,*T*] is the diagonal of $F{(P,{\theta}_{0})}^{-1}F({P}_{[0,t]},{\theta}_{0})$ (see Banks et al., 2008).

Now we can come back to the FP of EEG modeled by equations (1)–(5). In this case we cannot calculate the GSF corresponding to the PDE system that models the FP of EEG straightforward since instead of a dynamical system, in this case the independent variable is *x**G* *R*^{3}. The available data is now the value of the potential *u* at given points on the scalp *x _{j}* contaminated with noise and the observations are given by:

$$y({x}_{j})=u({x}_{j},\theta )+{e}_{j},\text{\hspace{1em}}{x}_{j}\in \Gamma ,\text{\hspace{1em}}j=1,\mathrm{\dots},n$$

(23)

where the observation errors *e _{j}*

In Rubio et al. (2009) it was proved that choosing an arbitrary order for the observation points *x _{j}*, the function:

$${\text{GSF}}_{\text{inc}}({x}_{j})={F}^{-1}\times \frac{{\nabla}_{q}h({x}_{j},\theta )}{{\sigma}^{2}({x}_{j})}\xb7{\nabla}_{\theta}h({x}_{j},\theta ),\text{\hspace{1em}}j=1,\mathrm{\dots},n.$$

(24)

is well-defined, that is, the value of GSF_{inc}(*x*) at a given observation point *x**G* does not depend on the order given to the observations points. Thus, we choose an order for the observation points on the scalp and perform the calculations.

We calculate the TSF and the GSF of *u* with respect to σ_{1} for a dipole source *J _{i}*(

In Figure Figure2,2, we show the absolute value of the TSF on the scalp for the same dipole location *r _{q}*= (0.3, 0.4, 0) and different dipoles moments. The stars indicate the positions of the scalp electrodes. In order to compare both sensitivity functions we have put an M at the position of the electrode where the two highest values of GSF

From the experiments we conclude that in this example TSF and GSF do not seem to provide the same information. It can be noted that in Figures Figures2B2B and D the maximum is not located at the electrode where TSF would suggest. Since the GSF is defined taking into account the influence that parameter variations have on the estimator $\widehat{\theta}$, theoretically it seems to be more suitable to look at the values of the GSF before performing the parameter estimation. More examples in different domains had to be performed and a theoretical analysis about the information provided of both sensitivity functions must be done. In future, based on these sensitivities, an optimal distribution of scalp electrodes may be determined in order to estimate the parameters of the model.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be constructed as a potential conflict of interest.

This research was supported in part by the Air Force Office of Scientific Research under grant FA9550-10-1-0037

- Banks H. T., Bihari K. L. (2001). Modeling and estimating uncertainty in parameter estimation. Inverse Probl. 17, 95–11110.1088/0266-5611/17/1/308 [Cross Ref]
- Banks H. T., Dediu S., Ernstberger S. L. (2008). Sensitivity functions and their uses in inverse problems. J. Inverse III-Posed Probl. 15, 683–708, ISSN (Online) 1569–3945
- Burns J. A., Lin T., Stanley L. (2005). A Petrov-Galerkin finite element method for interface problems arising in sensitivity computations. Comput. Math. Appl. 49, 1889–190310.1016/j.camwa.2004.06.037 [Cross Ref]
- El Badia A. (2000). Inverse source problem in an anisotropic medium by boundary measurements. Inverse Probl. 16, 651–66310.1088/0266-5611/16/3/308 [Cross Ref]
- Geddes L. A., Baker L. E. (1967). The specific resistant of biological materials – a compendium of data for the biomedical engineering and physiologist. Med. Biol. Eng. 5, 271–29310.1007/BF02474537 [PubMed] [Cross Ref]
- Hamalainen M., Hari R., Ilmoniemi R. J., Knuutila J., Lounasmaa O. (1993). Magnetoencephalography, theory, instrumentation and applications to noninvasive studies of the working human brain. Rev. Mod. Phys. 65, 414–48710.1103/RevModPhys.65.413 [Cross Ref]
- Kappel F., Batzel J. (2006). “Sensitivity analysis of a model of the cardiovascular system,” in Proceedings of the 28th IEEE EMBS Annual International Conference, New York [PubMed]
- Rubio D., Troparevsky M. I., Saintier N. (2009). “Generalized sensitivity analysis for an elliptic equation with interfaces,” in Proceedings of XIII Reunión de trabajo en procesamiento de la Información y control, RPIC 09, Rosario, Argentina
- Sarvas J. (1987). Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem. Phys. Med. Biol. 32, 11–2210.1088/0031-9155/32/1/004 [PubMed] [Cross Ref]
- Stanley L. G. (1999). Computational Methods for Sensitivity Analysis with Applications to Elliptic Boundary Value Problems. Ph.D. thesis, Department of Mathematics, Virginia Tech, Blacksburg, VA
- Thomaseth K., Cobelli C. (1999). Generalized sensitivity functions in physiological system identification. Ann. Biomed. Eng. 27, 607–61610.1114/1.207 [PubMed] [Cross Ref]
- Troparevsky M. I., Rubio D. (2003). On the weak solutions of the forward problem in EEG. J. Appl. Math. 12, 647–65610.1155/S1110757X03305030 [Cross Ref]
- Zhang Z. (1995). A fast method to compute surface potentials generated by dipoles within multilayer anisotropic spheres. Phys. Med. Biol. 40, 335–34910.1088/0031-9155/40/3/001 [PubMed] [Cross Ref]

Articles from Frontiers in Computational Neuroscience are provided here courtesy of **Frontiers Media SA**

PubMed Central Canada is a service of the Canadian Institutes of Health Research (CIHR) working in partnership with the National Research Council's national science library in cooperation with the National Center for Biotechnology Information at the U.S. National Library of Medicine(NCBI/NLM). It includes content provided to the PubMed Central International archive by participating publishers. |