E NERGY M ETABOLISM AND N EURONAL A CTIVITY : A P HYSIOLOGICAL M ODEL FOR B RAIN I MAGING Ana Rita Laceiras Gafaniz 2 de Dezembro de 2010 FACULDADE DE MEDICINA Universidade de Lisboa

I NTRODUCTION 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. 2

I NTRODUCTION : M ODEL D ESIGN FOR THE HRF 3 Na,K-ATPase

M OTIVATION A Physiologically-Based Haemodynamic linear model for the HRF ( Afonso et al (2007)) the Brain Group modulates the brain cells CMRO 2 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. 4

O BJECTIVES 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. 5

T HE N EURO -M ETABOLIC M ODEL : O VERVIEW 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 6

S ODIUM AND P OTASSIUM D YNAMICS : O RDINARY D IFFERENTIAL E QUATIONS Electrochemical gradient: Concentration gradient Na/K-Pump: Electric field Ion transport associated with the Electrical Activity

S ODIUM AND P OTASSIUM D YNAMICS : N EURONAL E LECTRICAL A CTIVITY Hodgkin-Huxley r(t) depolarisation hyperpolarisation repolarisation 8

S ODIUM AND P OTASSIUM D YNAMICS : T RANSFER F UNCTIONS 9

N EURONAL E LECTRICAL A CTIVITY AND ATP C ONSUMPTION 10 ATP Consumption Rate: ATP Consumption:

T HE M ITOCHONDRIA 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. 11

O VERALL N EURO -M ETABOLIC M ODEL : N EURONAL E LECTRICAL A CTIVITY AND ATP D YNAMICS 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 12

ATP D YNAMICS : T RANSFER F UNCTIONS 13

C OEFFICIENTS E STIMATION The model parameters were obtained from the literature, or estimated when they were not available Na(s) and K(s) coefficients ATP(s) coefficients 14

R ESULTS S USTAINED A CTIVATION AND R EPETITIVE A CTIVATION Comparison with the results published by Aubert & Costalat (2002) (in blue) Time constant Consistent with experimental work for mammalian CNS neurons 15

R ESULTS S USTAINED A CTIVATION AND R EPETITIVE A CTIVATION ATP dynamics: comparison with the results published by Aubert & Costalat (2002) (in blue) Time constant 16

P OLE -Z ERO (PZ) M AP PZ from the Na/K-ATPase: PZ from the mitochondria: 17

O VERALL T RANSFER F UNCTION p 1 mainly depends on ρ, the Na/K-ATPase activity time constant p 3 derives from the time constant, τ, for ATP production by the mitochondria Simplification using the Taylor Series Expansion 18

O VERALL T RANSFER F UNCTION 19

F REQUENCY RESPONSE Response to a 100Hz impulse train of spikes 20

C ONCLUSIONS AND F UTURE W ORK 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 21

R EFERENCES 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 IEEE International Conference of the Engineering in Medicine and Biology Society (EMBS), August 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, 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, M. F. Bear, B. W. Connors, and M. A. Paradiso, Neuroscience: Exploring the Brain. Williams & Wilkins, 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, M. D. Mann, “Control systems and homeostasis,” The Nervous System In Action, accessed at July 20, [Online]. Available: 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,

A KNOWLEDGEMENTS Prof. João Sanches Prof. Patrícia Figueiredo Prof. Fernando Lopes da Silva Prof. João Miranda Lemos Nuno Santos André Gomes David Afonso 23