García Grajales, Julián Andrés

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

Neurite: a finite difference continuum model of 
the electrophysiological-mechanical coupling in 
neurons under mechanical loading
Julián Andrés García Grajales
Advanced computing for science and engineering
Master thesis, UPM, CACI, Madrid, July 2012
Advisor: Prof. Jose Maria Peña
External advisor: Dr. Antoine Jerusalem
Outline
1. Introduction and objectives
2. Model. Neurite
3. Model analysis
4. Conclusions and future work
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1. Introduction and 
objective
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Adapted from Biophysics of computation. C. 
Kock. 1999
Extracellular fluid
Axon
• Neurons are excitable cells that transmit electrical and chemical 
signaling
• Axons are covered by layers of lipid and proteins (myelin)
• The myelin sheet is interrupted at regular intervals along axon by nodes 
(nodes of Ranvier)
• At nodes of Ranvier extracellular fluid gains direct access to the axonal 
membrane:
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Myelinated axons
- Regulate the flow of ions across 
membrane
- Are responsible for action potential 
creation and propagation
- Are described by Hodgkin Huxley model 
(sodium, potassium and leakage channels)
Mechanical traumas
- These injuries are the results of 
mechanical stresses/strains at 
generally high stress/strain-rates 
(tension, compression, shearing, 
etc.)
- Such damage can produce the 
disruption of axon functionalities, 
e.g. degradation of electrical 
properties 
- Ex vivo model of Shi and coworkers 
allows for quantification of axon 
electrical property loss after 
stretching and compression
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• Axonal injuries are one of the most common and 
devastating consequences of traumatic brain and spinal 
cord injury
J.M Jensen and R. Shi. Effects of 4-
Aminopyridine on stretched mammalian 
spinal cord: the role of potassium channels 
in axonal conduction. Journal of 
Neurophysiology. 2003; 90:  2334-2340
Objective
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• Cable theory and Hodgkin Huxley model to describe 
electrical conduction along axon (myelinated and nodes of 
Ranvier, respectively) 
• The mechanical model relates the electrical and 
mechanical properties
• In the application example, calibration and validation 
against experimental works of Shi and Whitebone (2006) 
and Ouyang et al (2010)
Develop an electro-mechanical model simulating the 
axonal electrical behavior during mechanical loading
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2. Model
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Mechanical and coupling models
Adapted from: A computational model coupling mechanics and electrophysiology in traumatic brain injury. Journal of Neurotrauma, 2012, Under review.
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Neurite
Without myelin for the nodes of Ranvier Independent of the strain
 Cable theory:
 Hodgkin and Huxley:
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Discretization
General scheme:
 Finite difference method (FDM)
 Spatial and temporal 
discretization
 Different type of elements
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Explicit scheme (1/2)
 Relates the current state of a variable to its and its 
neighbors old states
 Forward difference in time for first order derivative
 Numerically stable for a time step small enough
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Final set of equations for the explicit method:
IR-IR (pure IR):
NR-NR (pure NR):
IR-NR paranodal:
NR-IR paranodal:
Explicit scheme (2/2)
A  linear system of 
equations must be 
solved each time 
step
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Implicit scheme (1/2)
Relates the old stat of 
the variable and its 
and its neighbors 
current states
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IR-IR (pure IR):
NR-NR (pure NR):
IR-NR paranodal:
NR-IR paranodal:
Final set of equations for the implicit method:
Implicit scheme (1/2)
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Overall model
A computational model coupling mechanics and electrophysiology in traumatic brain injury. Journal of Neurotrauma, 2012, Under review.
4. Results and discussion
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Convergence = stability +consistency 
Stability:
Stability + consistency = convergence !!!
Consistency:
Unconditioned
Spatial
Explicit Implicit Unconditioned
Expectral analysis
Temporal
Guaranteed!!!
 Solutions calculated by the difference 
scheme at a given coordinate converge to 
its analytical PDE solution when the grid 
goes to zero (       ,          and      )
 Analytical solutions are unknown
 Consistency is assumed when the results 
do not change when the grid is decreased
Guaranteed!!!
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Electro-mechanical model
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Application example
Adapted from Shi and coworkers (2010) and 
Ouyang et al (2006)  
Calibration
Validation
from: A computational model coupling 
mechanics and electrophysiology in 
traumatic brain injury. Journal of 
Neurotrauma, 2012, Under review.
5. Conclusions and future work
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Conclusions and future work
Conclusions:
 Coupling Neurite to the mechanical model is a novelty in the field to 
simulate functional deficits produced by mechanical loading
 The development, testing and analysis of this new extension of Neurite
is the main contribution of this work
 This thesis constitutes a first step in the direction of a fully integrated 
mechanical-electrical-chemical in silico simulation platform for neurites
Future work:
 The electrical part of Neurite will be implemented in GPU’s 
(collaboration with DATSI department UPM).
 Application of Neurite to simulate action potential propagation in 
human nerves (Collaboration with Center of Biomedical Technology in 
UPM).
 Extent Neurite from FDM to FEM (3D)
 Coupling both parts of Neurite: the growth model + electro-
mechanical model
Thanks
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23
Backup slides
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Neuronal membrane
• Neuronal membrane is 
composed of:
– Lipid bilayer
– Ion channels 
• The membrane separates 
charges between inside and 
outside: acts as a capacitance C
• Current conductance along 
possible gates (passive ion 
channels, pores, etc.) is 
described by a resistance R
• Vrest represents the membrane 
potential at rest
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Biophysics of computation. C. Kock. 1999
J.A García
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Calibration (2/2)
• Looking for damage necessary:
Source at 730 µm 
away from the 
measuring point
Time (ms)
V
o
lt
s
 (
m
V
)
Neurons
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Franze and Guck. The biophysics of neuronal growth. Reports on 
Progress in Physics, 73:19pp, 2010
NEUROSCIENCE:
 Definition
 Computational 
neuroscience, 
Neuroinformatics, etc
 Applications
THE NEURON:
 Excitable cells
 Electrical and chemical signaling
 Axon and dendrites
 Networks
 Synapses
Extracellular fluid
Axon
• Axons are covered by layers of lipid and proteins (myelin)
• Myelin layers insulate axon from extracellular fluid
• The myelin sheet is interrupted at regular intervals along axon by 
nodes (nodes of Ranvier)
• At nodes of Ranvier extracellular fluid gains direct access to the 
axonal membrane
• The region where internodal part transforms to node of Ranvier is 
called paranodal region
• Passive behavior for myelinated part and active for the nodes of 
Ranvier
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Myelinated axons
Paranodal regions
• Neuronal membrane is composed of:
– Lipid bilayer
– Ion channels 
• Ion channels (uncovered at nodes of Ranvier):
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Membrane description
- Regulate the flow of ions across 
membrane
- Are responsible for action 
potential creation and propagation
- Are described by Hodgkin Huxley 
model (sodium, potassium and 
leakage channels) Adapted from Biophysics of computation. C. 
Kock. 1999
• Internodal part is described by cable theory
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Mechanical model
Adapted from: A computational model coupling 
mechanics and electrophysiology in traumatic 
brain injury. Journal of Neurotrauma, 2012, 
Under review.
Free parameters of the model:
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Coupling model
 The bridge between the mechanical 
model and Neurite
 A new parameter                      is 
proposed for distributing the strain along 
the axon
Adapted from: A computational model coupling 
mechanics and electrophysiology in traumatic 
brain injury. Journal of Neurotrauma, 2012, 
Under review.
and assuming incompressibility:
Conductances for Hodgkin and 
Huxley model do not change 
under mechanical  loading 


Neurite: a finite difference continuum model of the electrophysiological-mechanical coupling in neurons under mechanical loading

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