Transcript Energy Metabolism and Neuronal Activity: A Physiological

FACULDADE DE MEDICINA
Universidade de Lisboa
ENERGY METABOLISM AND
NEURONAL ACTIVITY:
A PHYSIOLOGICAL MODEL FOR
BRAIN IMAGING
Ana Rita Laceiras Gafaniz
2 de Dezembro de 2010
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
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SODIUM AND POTASSIUM DYNAMICS:
ORDINARY DIFFERENTIAL EQUATIONS
dNa
 Na   V  3 pump   Na r (t )
dt
dK
 K  V  2 pump   K r (t )
dt
Electrochemical gradient:
Concentration gradient
K 
Ke  K
d
Electric field
V 
Na 
Nae  Na
d
 V   ( Nae  Na  K e  K )

d
d
Na/K-Pump:
 pump  Na
Ion transport
associated with the
Electrical Activity
SODIUM AND POTASSIUM DYNAMICS:
NEURONAL ELECTRICAL ACTIVITY
r(t)
depolarisation
hyperpolarisation
Hodgkin-Huxley
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repolarisation
SODIUM AND POTASSIUM DYNAMICS:
TRANSFER FUNCTIONS
Na(s)  GN (s) Nae  GK (s) Ke  Gr ( s) R(s)
K ( s)  H N (s) Nae  H K (s) K e  H r (s) R(s)
1s   2
s 2  1s  2
3 s   4
GK ( s )  2
s   1s   2
5 s   6
Gr ( s)  2
s   1s   2
GN ( s ) 
1s   2
s 2  1s  2
 3s   4
H K (s)  2
s  1s   2
5s  6
H r (s)  2
s   1s   2
H N ( s) 
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NEURONAL ELECTRICAL ACTIVITY AND
ATP CONSUMPTION
ATP Consumption Rate:
ATPr (t )  Na(t )
ATPr ( s)  
5 s   6
R( s )
2
s  1s  2
Gr (s)
ATP Consumption:
ATPd ( s ) 
ATPr ( s )
s
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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
ATP(s)  LR (s) Ref (s)  LN (s) Nae (s)  LK (s) Ke (s)  Lr (s) R(s)
1s 2  2 s  3
LR ( s)  2
( s  1s  2 )(s 2  3 s  4 )
4 s 2  5 s  6
LN ( s )  2
( s  1s  2 )(s 2  3 s  4 )
7 s 2  8 s  9
LK ( s)  2
( s  1s  2 )(s 2  3 s  4 )
10 s 2  11s  12
Lr ( s )  2
( s  1s  2 )(s 2  3 s  4 )
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COEFFICIENTS ESTIMATION
Na(s) and K(s) coefficients
ATP(s) coefficients
The model parameters were obtained from the
literature, or estimated when they were not available
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RESULTS
SUSTAINED ACTIVATION AND REPETITIVE ACTIVATION
f  100Hz
f  230Hz
Comparison with the
results published by Aubert
& Costalat (2002) (in blue)
Time constant
 pump  33s
Consistent with
experimental work for
mammalian CNS neurons
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RESULTS
SUSTAINED ACTIVATION AND REPETITIVE ACTIVATION
Time constant
  30 s
ATP dynamics: comparison with the results published by
Aubert & Costalat (2002) (in blue)
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POLE-ZERO (PZ) MAP
PZ from the Na/K-ATPase:
p1  0.031rad s 1
p2  0.653rad s 1
z1  0.650rad s 1
PZ from the mitochondria:
p3  0.033rad s 1
p4  30.00rad s 1
z2  30.03rad s 1
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OVERALL TRANSFER FUNCTION
0
12 p1 p3
Lr ( s) 
, 0 
( s  p1 )(s  p3 )
 2 4
 1   12  4 2
p1 
2
    2  4
p3 
2


p1 
Simplification using the
Taylor Series Expansion
 2
1

p3 

 3  
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
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OVERALL TRANSFER FUNCTION
p1  p3

12 p 2
Lr ( s ) 
, 0 
,
2
( s  p)
 2 4
p
p1  p3
 0.0320rads1
2
3
6
ATP( s) 
Ref ( s) 
Nae ( s) 
 2 4
 2 4
9
12 p 2
1
K e ( s) 
R( s )
2
 2 4
 2 4 ( s  p)
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FREQUENCY RESPONSE
c  0.032rads1
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Response to a 100Hz impulse train of spikes
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
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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
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J. Malmivuo and R. Plonsey, Bioelectromagnetism, Principles and Applications of
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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
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