Institute of Mathematics and Image Computing, University of Lübeck, Ratzeburger Allee 160, 23538 Lübeck, Germany

Department of Psychiatry and Psychotherapy, Division of Psychoneurobiology, University of Lübeck, Ratzeburger Allee 160, 23538 Lübeck, Germany

Department of Mathematics, Virginia Tech, 225 Stanger Street, 474 McBryde Hall, Blacksburg, VA 24061, USA

Abstract

Background

Energy homeostasis ensures the functionality of the entire organism. The human brain as a missing link in the global regulation of the complex whole body energy metabolism is subject to recent investigation. The goal of this study is to gain insight into the influence of neuronal brain activity on cerebral and peripheral energy metabolism. In particular, the tight link between brain energy supply and metabolic responses of the organism is of interest. We aim to identifying regulatory elements of the human brain in the whole body energy homeostasis.

Methods

First, we introduce a general mathematical model describing the human whole body energy metabolism. It takes into account the two central roles of the brain in terms of energy metabolism. The brain is considered as energy consumer as well as regulatory instance. Secondly, we validate our mathematical model by experimental data. Cerebral high-energy phosphate content and peripheral glucose metabolism are measured in healthy men upon neuronal activation induced by transcranial direct current stimulation versus sham stimulation. By parameter estimation we identify model parameters that provide insight into underlying neurophysiological processes. Identified parameters reveal effects of neuronal activity on regulatory mechanisms of systemic glucose metabolism.

Results

Our examinations support the view that the brain increases its glucose supply upon neuronal activation. The results indicate that the brain supplies itself with energy according to its needs, and preeminence of cerebral energy supply is reflected. This mechanism ensures balanced cerebral energy homeostasis.

Conclusions

The hypothesis of the central role of the brain in whole body energy homeostasis as active controller is supported.

Background

Regulation of the energy metabolism is crucial to ensure functionality of the human organism. However, the interactions of numerous energy metabolites and neuroendocrine mechanisms in the complex regulation are still not completely understood.

So far, there exist various theoretical approaches to explain the regulation of the human energy metabolism. Two traditional concepts are called the glucostatic theory

The “Selfish Brain Theory” provides a new approach to explain the regulation of the human whole body energy metabolism

Hence, identifying control mechanisms of the brain energy homeostasis is a major goal in obtaining a systemic understanding of the human overall energy metabolism and thereby providing insight into pathological regulation

In the present study, we aim to gain specific information about the regulatory elements of the human brain in the systemic energy homeostasis. Therefore, we combine mathematical modeling and experimental data. In our novel approach, the integrative behavior of the human whole body energy metabolism is mathematically modeled in a compact dynamical system

A close relation between brain energy homeostasis and systemic glucose metabolism has been suggested several times, e.g.,

In the following Material and methods section, we describe the experimental study in Experimental study section and introduce the mathematical model in Brain-centered energy metabolism model section. Parameter estimation methods are presented in Parameter identification section and Parameter identification setup section. Results of the parameter identification are investigated in Results section. We close with a discussion and a brief outlook in Conclusions section.

Material and methods

Experimental study

The goal of our experimental study is to investigate the close link between neuronal brain activity and subsequent metabolic responses of the human organism at a systemic level. In

Design of the experimental study

**Design of the experimental study.** Blood samples and ^{31}phosphorus magnetic resonance spectra (MRS) were regularly taken under steady state conditions of an euglycemic hyperinsulinemic clamp (EHC) to monitor cerebral and peripheral energy metabolism. Transcranial direct current stimulation (tDCS) or sham stimulation, respectively, were induced to affect neuronal brain activity.

In a randomized sham-controlled crossover design, a homogeneous group of 15 healthy young male volunteers with a body mass index of 23.2 ± 0.38 kg/m^{2} is examined. Neuronal brain activity is stimulated by transcranial direct current stimulation (tDCS) during the time interval

^{31}Phosphorus magnetic resonance spectroscopy (^{31}P-MRS) allows performing non-invasive in vivo measurements of brain metabolites that are centrally involved in the energy metabolism. Phosphate metabolites such as adenosinetriphosphate (ATP), i.e., the sum of α-, β-, and γ-ATP, as well as phosphocreatine are measured in the cortex reflecting the overall high-energy phosphate turnover ^{31}P-MR spectra are measured at times

During euglycemic-hyperinsulinemic clamping, an insulin infusion at the predetermined fixed dosage of 1.5 mU (kg min)^{−1} and a variable glucose infusion are administered in order to reach stable plasma glucose concentrations between 4.5 and 5.5 mmol/l. Under steady-state conditions of euglycemia, the glucose infusion rate equals glucose uptake by all tissues in the body

While blood glucose and insulin concentrations do not differ between conditions, overall cerebral high-energy phosphate measurements display a biphasic course upon tDCS as compared with the sham condition, see ^{31}P-MR spectra reveal a rapid increase above the control condition followed by a return of the ATP/Pi ratio to baseline levels. Glucose infusion rates show the same biphasic response to tDCS indicating that transcranial stimulation improves systemic glucose tolerance

Experimental data

**Experimental data.** Effects of transcranial direct current stimulation (red) on cerebral ATP/Pi, blood glucose, and insulin during a hyperinsulinemic-euglycemic glucose clamp condition as compared with the sham stimulation (black). The top right figure shows relative changes of cerebral ATP/Pi in proportion to matched sham condition values. Data show a biphasic course with an initial drop by trend followed by a rise. The gray background indicates the stimulation interval. Data represent mean values +/- standard error of mean (Figure reproduced from

The experimental data demonstrate that transcranial brain stimulation not only evokes alterations in local neuronal processes but also clearly influences brain energy homeostasis and peripheral metabolic systems regulated by the brain

Nevertheless, the mechanisms underlying these experimental observations remain unknown. Concerning the specific mechanisms by which neuronal excitation, via a drop in high-energy phosphate content, improves glucose tolerance, one can only speculate at this point.

The objective of the following mathematical analysis is to gain insight into physiological mechanisms underlying the effects of brain stimulation on cerebral and peripheral energy metabolism. In order to clarify the underlying mechanisms in this context, we combine experimental data with the mathematical model introduced in the following subsection.

Brain-centered energy metabolism model

Physiological processes may be described by systems of ordinary differential equations,

where _{0} denotes the initial conditions. We collect unknown model parameters to be estimated in the vector _{1}, …, _{
m
})^{⊤}. In the following, given parameters regarded as constant are assembled in a vector

A collection of mathematical models of this kind describing interactions of main components of the human glucose metabolism can found in the book of Chee and Fernando

Here, we regard a mathematical model of the human whole body energy metabolism considering the brain not only as energy consumer but more importantly as a superordinate controller, compare

Illustration of the brain centered energy model (2)

**Illustration of the brain centered energy model (2).** Energy fluxes between compartments (solid) and control signals directing energy fluxes within the organism (dashed).

The brain-centered model of the energy metabolism is given by the system of four ordinary differential equations,

Cerebral energy content, i.e., high-energy phosphates in the brain, is denoted by

Conceptually, our model bases on conservation of energy. In general, stimulatory influences are modeled as proportional relations and inversely proportional relations describe prohibitive influences. The glucose flux from the blood into the brain crossing the blood brain barrier is proportional to intensified by a factor _{1} [M s^{−1}]. This factor quantifies the glucose flux rate from the blood into the brain across the blood brain barrier. Suppose the cerebral energy content

Our model combines energy resources and metabolites, such as glycogen, glucose and lactate, in the compartment _{2} [M s^{−1}], i.e., _{2}

The hormone insulin acts not only as local response to the blood glucose concentration. Moreover, it is regarded as central feedback signal of the brain with an insulin secretion factor _{3} [s^{−1}]. Notice that with low cerebral energy, ventromedial hypothalamic centers inhibit pancreatic insulin secretion _{4} [M s^{−1}] and by _{5} [M s^{−1}].

Insulin-dependent glucose uptake from the blood into the energy resources compartment is modeled by _{1}
_{1} [(M s)^{−1}]. This flux mainly comprises glucose uptake into the peripheral stores, i.e., muscle and fat tissue. To accelerate this flux, glucose and insulin need to be available in the blood at the same time in order to activate glucose uptake via the insulin-dependent glucose transporter GLUT4.

Degradation of insulin is supposed to be of first order with the insulin degradation rate _{2} [s^{−1}]. External glucose infusion is denoted by _{
ext
}(^{−1}], insulin infusion is _{
ext
}(^{−1}].

To meet the simplified notation from Equation (1) we collect the state variables ^{⊤} and the parameters _{1}, …, _{5})^{⊤}. The constants _{1}, _{2})^{⊤} and time-dependent external infusions _{
ext
}, _{
ext
} are not explicitly shown in (1). Notice that all parameters, constants, and states are non-negative. Model properties he been investigated in detail and it has been shown that the model realistically reproduces qualitative and quantitative behavior of the whole body energy metabolism even for a large class of physiological interventions (see

To accommodate the characteristics of the experimental study, the dynamical system (2) slightly differs from the model introduced in

Parameter identification

In the following, we introduce the general technique to estimate the model parameters _{1}, …, _{5})^{⊤} of our dynamical system in Brain-centered energy metabolism model section using experimental data presented in Experimental study section.

In general, parameter identification problems for ordinary differential equations can be stated as follows

Equation (3) states a classical constrained optimization problem, where constraints are given by an initial value problem _{2} denoting the Euclidian norm, we minimize the distance between the model solution at the times _{1}, …, _{
k
})^{⊤}, where the data are measured and given data

We seek to find

Here, we follow an approach similar to methods proposed by Ramsay et al.

The equivalence of optimization problem (3) and (4) stays true for any appropriate integral norm ∙ (here we choose the ^{2} –norm). Constrained optimization problems such as (4) are commonly approximated by performing a Lagrangian relaxation

with the Lagrangian multiplier _{1}, …, _{
k
} and _{
t
} be a finite differences operator approximating _{1}, …, _{
k
}, then we can restate the optimization problem (5) as the discretized and unconstrained optimization problem.

Notice that we can neglect the remaining constraint in (5) since _{0} = _{1} is already included in the search parameters and is therefore always fulfilled. Equation (6) describes the general parameter estimation framework. This optimization problem has the advantages that it is robust and the unconstrained nature allows to use fast gradient-based methods, for details see _{1}, …, _{
k
} are t dense, one may want to utilize a spline function

One further choice to make is choosing the regularization term

where

Parameter identification setup

Next, we present the parameter estimation setup for the model in Section 2.2 with the given data from Section 2.1. As derived in Section 2.2, all parameters of our model (2) have a physiological interpretation. Since insulin is at hyperphysiological levels in our experimental examination we will consider the dependent parameters _{1} and _{2} to be constant. Flakoll PJ, Wentzel LS, Rice DE, Hill JO and Abumrad NN _{1} ≈ 0.06 (pM min)^{−1}. Information about insulin clearance is given by _{2} ≈ 1.4 min^{−1}. This yields the fixed parameter values _{1,}
_{2}) = (0.06, 1.4)^{⊤}.

For the parameter ^{−1}, see

In order to establish global convergence, we choose a Monte Carlo sampling technique of the initial guess _{0}. We pick 500 randomly chosen normally distributed samples with mean

To solve the optimization problem (4) numerically we use a Gauss-Newton method with Armijo line search, see ^{−3} and ^{−7} for our numerical investigations. By empirical observations these values lead to a good balance between under- and overfitting.

Results

Model validation

The goal of this section is to show that parameter values of our dynamical system (2) can be chosen such that the resulting mathematical model is able to approximate the experimental measurements of the study described in Section 2.1. To validate the mathematical model, we first identify the model parameters by mean values of the experimental data for cerebral ATP/Pi, blood glucose, and insulin concentration in the sham stimulation condition. Note, for the energy resources compartment no experimental data are available, and we arbitrarily set ^{2} = 1000 be fairly high, which allows the optimization method to pick adequately different values for ^{−5} for the placebo group.

In order to test the accuracy of the estimation, we calculate the distance between data and calculated model by the relative mean squared error

where ^{−3}.

Experimental data (error bars) versus mathematical model (solid lines) for the mean sham stimulation data

**Experimental data (error bars) versus mathematical model (solid lines) for the mean sham stimulation data.** Error bars show mean values and respective standard errors of the mean of cerebral ATP/Pi, blood glucose, and insulin levels. For energy resources, no experimental data are available. Solid lines represent model predictions with the estimated parameter values ^{−5} and relative mean squared error ^{−3}.

The parameter estimates for the mean data from the tDCS intervention read ^{−4} and ^{−3} and are illustrated in Figure

Experimental data (error bars) versus mathematical model (solid lines) for the mean tDCS stimulation data

**Experimental data (error bars) versus mathematical model (solid lines) for the mean tDCS stimulation data.** Error bars show mean values and respective standard errors of the mean of cerebral ATP/Pi, blood glucose, and insulin levels. For energy resources, no experimental data are available. Solid lines represent model predictions with the estimated parameter values ^{−4} and relative mean squared error ^{−3}.

Since we conduct parameter identification with 500 random initial values _{0} as well as the values

With the estimated parameter values ^{⊤} compared with experimental data for sham and tDCS interventions. The model predictions closely approximate the measurements. Ideally a glycemic-hyperinsulinemic clamp generates constant blood glucose and plasma insulin levels. Constant glucose and insulin levels by themselves do not contribute much insight to the parameter estimation. However, the glucose and insulin fluxes _{
ext
} and _{
ext
} are recorded and are included in the parameter estimation. Note that blood glucose predictions are slightly unsteady due to the external glucose infusion.

To validate the mathematical model, we perform a sensitivity analysis. The results of the sensitivity analysis for the two central parameters _{1} and _{4} are illustrated in Figures _{2}, _{3,}
_{5} and initial values _{0} also affects the predicted profiles, indicating their sensitivity to changes in these parameters as well (analysis not shown). Energy resources _{5}.

Effects of parameter modifications on model predictions of cerebral ATP, blood glucose, and insulin concentrations as well as on energy resources

**Effects of parameter modifications on model predictions of cerebral ATP, blood glucose, and insulin concentrations as well as on energy resources.** Stepwise increase in cerebral glucose uptake p_{1} by 150, 200, and 250% leads to marked elevations of predicted brain ATP and blood insulin as well as reduced blood glucose profiles. Hence, the predictions are very sensitive to changes in this parameter.

Modifying cerebral energy consumption p_{4} affects the predicted profiles of cerebral ATP, blood glucose, and insulin concentrations in a sensitive manner

**Modifying cerebral energy consumption p**
_{
4
}
**affects the predicted profiles of cerebral ATP, blood glucose, and insulin concentrations in a sensitive manner.**

Effects of tDCS on underlying physiological mechanisms

While Section 3.2 considers the population of placebo and tDCS data, we next investigate each individual data set using the same settings as before. We identify the parameters

Box-and-whisker diagrams of identified parameter values for current and sham stimulation, respectively

**Box-and-whisker diagrams of identified parameter values for current and sham stimulation, respectively.**

We statistically investigate the effects of tDCS on estimated parameter values. Each model parameter has a physiological interpretation. Thereby, our examinations provide insight into underlying physiological mechanisms that are not measurable in the experimental study.

We identify outliers _{1} and above quartile _{3} by

**
sham
**

**
tDCS
**

0.0555 ± 0.0709

0.1362 ± 0.1752

0.0785 ± 0.0463

0.0583 ± 0.0559

11.6895 ± 3.5051

10.4485 ± 2.2179

0.0320 ± 0.0380

0.0722 ± 0.0896

0.7983 ± 0.1320

0.8018 ± 0.2702

In order to investigate the effects of tDCS on underlying physiological mechanisms in the systemic energy metabolism, we compare mean estimated parameter values for tDCS and sham stimulation. Identified parameter ratios between tDCS and sham condition are shown and statistically significant values are highlighted in Table

^{a}Significance, P = 0.04, ^{b}Significance, P = 0.08, and ^{c}Significance, P = 0.03. Statistical analyses were performed by analysis of variance to identify main effects in experimental condition comparisons.

**tDCS/sham**

2.45^{a}

0.74^{b}

0.89

2.26^{c}

1.00

Glucose flux

The physiological interpretation of the obtained ratios reads as follows: TDCS excites cortical neurons. Our results verify that this leads to significantly increased cerebral energy consumption

Moreover, we analyze cerebral energy supply on demand. Therefore, we calculate the ratio of energy flux

Insulin production rate

Our results are in line with the hypothesis that the brain supplies itself with glucose in dependence of its own needs

Our findings explain the experimentally observed biphasic effect of tDCS on cerebral energy content, compare

Additionally, we find a significant decrease in energy flux

Peripheral energy consumption

Conclusions

For the first time, the relationship between neuronal brain activity and systemic energy metabolism was investigated in the experimental study

Modeling approaches, especially in physiological applications, feature limitations in their compactness and in validity of the model functions. Energy homeostasis comprises a tremendous number of metabolites and complex physiological interrelations that are sufficiently relevant to be considered in the model equations. Moreover, most of these mechanisms are not yet sufficiently quantified to be used in mathematical models. To account for these limitations, we restricted our mathematical model to only include widely accepted and fundamental physiological relations. We showed that the whole-body energy metabolism can be realistically modeled and experimental data are reasonably predicted. Therefore, limited but relevant conclusions can be drawn from our findings. Nevertheless, in future work we aim to include further model refinements.

We are able to identify parameter values, for which our mathematical model reproduces experimentally acquired data, see Section 3.1. Our results verify the presented physiological mechanisms and validate the mathematical model. We for the first time developed a mathematical model predicting experimental data of cerebral and peripheral metabolites at the same time.

As mentioned above, the parameters of our mathematical model have a physiological interpretation. Analyzing the identified model parameters thus allow to draw conclusions about physiological mechanisms underlying the experimental data, see Effects of tDCS on underlying physiological mechanisms section. Thereby, we are able to explain effects in the experimental observations.

We experimentally observe a decrease followed by an increase in cerebral high-energy phosphates upon tDCS due to neuronal activation. However, underlying physiological mechanisms explaining this experimental findings remain unknown at this point. By the validity of our parameter estimation method we can draw the following conclusions: Firstly, cerebral energy consumption significantly increases upon tDCS compared with sham stimulation. This explains the initial drop in brain energy level. Secondly, our findings reflect a significant increase in glucose transport rate across the blood brain barrier. Thirdly, we observe an improvement of the allocation mechanism upon stimulation, which is expressed by an increase in the ratio

In forthcoming investigations, we plan to investigate pathological conditions caused by deregulations in the energy metabolism such as obesity and diabetes mellitus. We want to identify model parameters of pathologic states thereby shedding light on defects causing metabolic diseases.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

BG conceived and conducted the mathematical analyses, interpreted the results, wrote and edited the manuscript, and approved the final version for submission; KMO conducted the experimental study, interpreted the results, edited the manuscript, and approved the final version for submission; MC conceived and conducted the mathematical analyses, interpreted the results, edited the manuscript, and approved the final version for submission. All authors read and approved the final manuscript.

Acknowledgement

This work was supported by the Graduate School for Computing in Medicine and Life Sciences funded by the German Research Foundation [DFG GSC 235/1] and by the Federal Ministry of Education and Research (BMBF).