ENERGY METABOLISM AND NEURONAL ACTIVITY: A PHYSIOLOGICAL MODEL FOR BRAIN IMAGING

Ana Rita Laceiras Gafaniz2 de Dezembro de 2010

FACULDADE DE MEDICINA

Universidade de Lisboa

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INTRODUCTION Functional Magnetic Resonance Imaging (fMRI)

is a widely used method to detect the activated brain regions due to a stimulus application.

The Blood-Oxygenation-Level-Dependent (BOLD) signal is based on the well-established correlation between neuronal activity, energy metabolism and haemodynamics.

The BOLD effect is small and data is noisy, turning this inference problem a difficult task

An accurate knowledge of the Haemodynamic Response Function (HRF) to a localized neural stimulus is critical, in order to interpret the fMRI data confidently.

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INTRODUCTION: MODEL DESIGN FOR THE HRF

Na,K-ATPase

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MOTIVATION A Physiologically-Based Haemodynamic linear model for

the HRF (Afonso et al (2007))

the Brain Group modulates the brain cells CMRO2 and the vascular demand;

the Vessel Group modulates the summed effect of CBV and CBF vascular changes on the oxyHb/deoxyHb rate in and around blood vessels;

the Control Group for the systemic negative feedback control over vasodilation.

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OBJECTIVES

Obtain a practical, tractable and simultaneously accurate mathematical model to describe the neuro-metabolic and neuro-vascular couplings that lead to the BOLD effect.

A physiologically-based lineal model describing the relation between the neuronal electrical activity and the ATP dynamics is proposed.

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THE NEURO-METABOLIC MODEL:OVERVIEW

a) Na/K-ATPase; b) K+ leak channels; c) Na+ leak channels; d) Na+ Voltage Gated Channels; e) K+ Voltage Gated Channels; f) Mitochondria g) Cellular Membrane

SODIUM AND POTASSIUM DYNAMICS:ORDINARY DIFFERENTIAL EQUATIONS

Electrochemical gradient:Concentration gradient

Na/K-Pump:

dNaNaNa e

dKKK e

Electric field

dKKNaNa

dVV ee )(

Napump Ion transport

associated with theElectrical Activity

)(2

)(3

trVKdtdK

trVNadtdNa

Kpump

Napump

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SODIUM AND POTASSIUM DYNAMICS:NEURONAL ELECTRICAL ACTIVITY

Hodgkin-Huxley

r(t)

depolarisation

hyperpolarisation

repolarisation

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SODIUM AND POTASSIUM DYNAMICS:TRANSFER FUNCTIONS

)()()()()()()()()()(sRsHKsHNasHsKsRsGKsGNasGsNa

reKeN

reKeN

212

65

212

43

212

21

)(

)(

)(

ssssG

ssssG

ssssG

r

K

N

212

65

212

43

212

21

)(

)(

)(

ssssH

ssssH

ssssH

r

K

N

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NEURONAL ELECTRICAL ACTIVITY AND ATP CONSUMPTION

)(sGr

)()(21

265 sR

ssssATPr

ATP Consumption Rate:

ATP Consumption:

ssATPsATP r

d)()(

)()( tNatATPr

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THE MITOCHONDRIA

The mitochondria acts as a regulator, from a Control Theory perspective

With a type-I system, the steady-state error to the step is zero and it is finite to the ramp.

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OVERALL NEURO-METABOLIC MODEL:NEURONAL ELECTRICAL ACTIVITY AND ATP DYNAMICS

The dynamic evolution of the intracellular concentration of ATP along the time results from the contribution of the ATP consumption, due to the Na,K-ATPase activity the ATP synthesis, by the mitochondrial activity

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ATP DYNAMICS:TRANSFER FUNCTIONS

)()()()()()()()()( sRsLsKsLsNasLsRefsLsATP reKeNR

))(()(

))(()(

))(()(

))(()(

432

212

12112

10

432

212

982

7

432

212

652

4

432

212

322

1

sssssssL

sssssssL

sssssssL

sssssssL

r

K

N

R

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COEFFICIENTS ESTIMATION

The model parameters were obtained from the literature, or estimated when they were not available

Na(s) and K(s) coefficients

ATP(s) coefficients

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RESULTS SUSTAINED ACTIVATION AND REPETITIVE ACTIVATION

Comparison with the results published by Aubert & Costalat (2002) (in blue)Time constant

spump 33Consistent with experimental work for mammalian CNS neurons

Hzf 100 Hzf 230

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RESULTS SUSTAINED ACTIVATION AND REPETITIVE ACTIVATION

ATP dynamics: comparison with the results published by Aubert & Costalat (2002) (in blue)

Time constants30

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POLE-ZERO (PZ) MAP

PZ from the Na/K-ATPase:

PZ from the mitochondria:

11 031.0 sradp

12 653.0 sradp

11 650.0 sradz

13 033.0 sradp

14 00.30 sradp

12 03.30 sradz

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OVERALL TRANSFER FUNCTION

p1 mainly depends on ρ, the Na/K-ATPase activity time constant

p3 derives from the time constant, τ, for ATP production by the mitochondria

42

31120

31

0 ,))((

)(

pppsps

sLr

24

24

2

3

2211

1

p

p

3

1

21 3

p

pSimplification using

the Taylor Series Expansion

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OVERALL TRANSFER FUNCTION

31 pp

131

42

212

02

0320.02

,,)(

)(

radsppp

pps

sLr

)()(

1)(

)()()(

242

212

42

9

42

6

42

3

sRps

psK

sNasRefsATP

e

e

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FREQUENCY RESPONSE

Response to a 100Hz impulse train of spikes

1032.0 radsc

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CONCLUSIONS AND FUTURE WORK A physiologically-based model

representing the ATP dynamics as a function of the neuronal electrical activity was proposed

A second order linear system with no zeros

Model parameters tuned with data obtained from the literature.

Validation with real data Incorporate the Neuro-Metabolic Model in

a more general model describing the Haemodynamic Response Function

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REFERENCES D. M. Afonso, J. M. Sanches, and M. H. Lauterbach, “Neural physiological modeling

towards a hemodynamic response function for fMRI,” in 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, EMBC 2007. IEEE International Conference of the Engineering in Medicine and Biology Society (EMBS), August 2007.

A. Aubert and R. Costalat, “A model of the coupling between electrical activity, metabolism, and hemodynamics: Application to the interpretation of functional neuroimaging,” Neuroimage, vol. 17, pp. 1162–1181, 2002.

S. Ogawa, T. M. Lee, A. R. Kay, and D.W. Tank, “Brain magnetic resonance imaging with contrast dependent on blood oxygenation,” in Proceedings of the National Academy of Sciences, S. U. H. Press, Ed., vol. 87. National Academy of Sciences, September 1990, pp. 9868–9872.

J. Malmivuo and R. Plonsey, Bioelectromagnetism, Principles and Applications of Bioelectric and Biomagnetic Fields. Oxford University Press, 1995.

M. F. Bear, B. W. Connors, and M. A. Paradiso, Neuroscience: Exploring the Brain. Williams & Wilkins, 1996.

A. L. Hodgkin and A. F. Huxley, “A quantitative description of membrane current and and its application to conduction and excitation in nerve,” Journal of Physiology (London), vol. 117, pp. 500–544, 1952.

M. D. Mann, “Control systems and homeostasis,” The Nervous System In Action, accessed at July 20, 2010. [Online]. Available: http://www.unmc.edu/physiology/Mann/mann2.html

D. Attwell and S. B. Laughlin, “An energy budget for signaling in the grey matter of the brain,” Journal of Cerebral Blood Flow & Metabolism, vol. 21, pp. 1133–1145, 2001.

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AKNOWLEDGEMENTS

Prof. João Sanches Prof. Patrícia Figueiredo Prof. Fernando Lopes da Silva Prof. João Miranda Lemos Nuno Santos André Gomes David Afonso