Propagation of Signals Along Nonuniform Axon

1
Propagation of Signals Along Non-uniform Axon

• STEPHEN CLARK
• YEVGENIY GOKUN
• HSIU-TSUN HSIEH

PRASANNA KARUNANAYAKA NAMYONG LEE MIKE MARTIN
2
Todays Outline

• Neural Signal Propagation
• History, Motivation
• Disease, Pathologies
• Mathematics
• HH, FN, ML Models
• Computation
• Numerics
• Simulations
• Conjectures
• Parameter Ranges
• Biological Relevance

3
Neural Dynamics

• Neural Signal Propagation
• History Motivation
• Action Potentials Dendritic Conduction
• Branching Patterns Traveling Waves
• Applications
• Disease, Amyotrophic Lateral Sclerosis (ALS)
• Myelination
• Cardiac Propagation
• Pain Mechanism

4
Basic Axonal Element Propagation

• The axonal current is predominantly made up of
  ionic flow (not electrons) and its direction is
  not longitudinal but transverse into the cell

5
Potential Concerns

• Recall,
• In squid giant axon, the principal ionic currents
  used to maintain potential are that of Na and
  K
• We lump other ionic currents into a leakage
  current
• Choice of linear I-V curves for the three
  different channel types is motivated largely by
  experimental data. Other species may have other
  I-V curves, but the results here are claimed to
  be qualitatively correct all the same

6
Is Something Amiss?

• We can write,
• where and
• Solving the above differential equation reveals a
  time constant on the order of 1 msec given by
• With a steady applied current, the membrane
  potential should equilibrate quickly to
• For sufficiently small applied currents, this is
  the case however, for larger currents it is not.
  Assuming the model is correct, the only
  explanation is that the conductances are not
  constant but instead depend on the potential.

7
The Hodgkin-Huxley Equations

• The equations for the space clamped axon are
• where

8
Hodgkin-Huxley Equations contd

• For the sodium and potassium channel dynamics,
  Hodgkin Huxley used

Sir Bernard Katz 1911-2003 Nobel Prize 1970
PICTURE SOURCE www.nobel.se
9
Hodgkin Huxley Action Potential
Simulation of the superthreshold response of the
axon to a strong depolarizing (I 10.0 mA/cm2)
current pulse of 5 msec duration.

• Four phases
• Upstroke
• Excited
• Refractory
• Recovery

10
Gating During an Action Potential
Simulation of the superthreshold response of the
axon to a strong depolarizing(I 10.0 mA/cm2)
current pulse of 5 msec duration.
Na Activation
Na Inactivation
h(t)
K Activation
n(t)
m(t)
11
Conductance during an Action Potential
Simulation of the superthreshold response of the
axon to a strong depolarizing (I 10.0 mA/cm2)
current pulse of 5 msec duration.
Na Conductance
gNa
K Conductance
gK
12
Whos on First?
Note the small time constant for Na activation
and recall that the potential responds quickly to
a rapid Na increase
Na Activation
Na Inactivation
K Activation
tn(v)
th(v)
tm(v)
13
Steady-state Functions
h8(v)
Na Activation
m8(v)
n8(v)
Na Inactivation
K Activation
ve gt vr implies h8(ve) lt h8(vr) and n8(ve) gt
n8(vr) So,while v is at the excited state, h
begins to decrease (inactivating Na
conductance), and n starts to increase
(activating the K conductance).
14
Gating Symmetry
Notice the symmetry between the Na Inactivation
and the K Activation In his reduction, FitzHugh
assumed that h n 0.8
Na Activation
Na Inactivation
h(t)
K Activation
n(t)
m(t)
15
Pathologies

• Neural Signal Propagation
• History Motivation
• Applications
• Cardiac propagation
• Depressed Excitability
• Echo in Purkinje Fibers
• Different diffusivity
• Not thickness, but sickness
• Disease, ALS, Myelination
• Possibly Alzheimers Parkinsons, too

16
Amyotrophic Lateral Sclerosis

• Collateral sprouting in patient with ALS
• Terminal branches from a single axon innervate
  several neuromuscular junctions

PICTURE SOURCE www.lougehrigsdisease.net
17
Amyotrophic Lateral Sclerosis
Collateral Sprouting
PICTURE SOURCE www.lougehrigsdisease.net
18
Amyotrophic Lateral Sclerosis
Terminal loss of myelin on an atrophic axon
PICTURE SOURCE www.lougehrigsdisease.net
19
Amyotrophic Lateral Sclerosis
Video courtesy of Anthony Brown, OSU

• Progressive Neurodegenerative Disease
• Neurofilaments accumulate at the Nodes of Ranvier
• Related to abnormal protein folding along
  cytoskeletal transport
• Nodes typically larger than myelinated axon, but
  more for ALS

20
ALS Lou Gehrigs Disease
Lou Gehrigs Farewell Address 4 July 1939 NY, NY
Lou Gehrig 1904-1941
PICTURE SOURCE www.lougehrigsdisease.net
21
Modeling Dendrities Historical background
22
Cable Equation
Ohms Law
Kirchoffs Law
Discrete
Continuous
The cable equation
23
LinearCable Theory
24
Mathematical Modeling

• Mathematical Model Equations
• Bistable Equation
• FitzHugh-Nagumo (1961-62)
• Qualitative, but representative
• Morris-Lecar (1981)
• Physiologically motivated
• Hodgkin-Huxley (1952)

25
Bistable Equation

• Traveling Wave Solution
• solution that propagates with constant shape and
  velocity
• Existence of Traveling Wave

26
Fitzhugh-Nagumo Equation

• Existence/Uniqueness/Stability of Traveling Wave
• both pulse type transition type traveling waves
• How to construct a traveling pulse?
• Let use geometric singular
  perturbation theory

27
How to construct a traveling pulse?

• Let
• Find the velocity c by using the shooting
  method
• Find the heteroclinic orbit such that

28
Singular Homoclinic Orbit

29
Pieces of Singular Wave Solution

• Silent and Active Phases
• Introduce slow time scale set
• The singular solution lies along the cubic curve
• Find such that

30
Singular Homoclinic Orbit

IMAGE SOURCE David Termans MBI Notes
31
Why Blocking Happened (Mathematical understanding)

• Use comparison theorem
• Pauwelussen 82

Upper Solution
Traveling Wave
32
Morris-Lecar Equation (1981)

• Quantitatively accurate model of
  neurophysiological process
• Parameters are determined from careful
  experimental data
• More realistic than Fitzhugh-Nagumo but has only
  two independent variables accounts for specific
  ion channels in barnacle muscle fiber
• Existence of traveling wave solution

33
The Hodgkin-Huxley Equations

• The equations for the space clamped axon are
• for the sodium and potassium channels, Hodgkin
  Huxley used

34
Note on Numerical Methods(Method of Lines)

• System of PDE ODEs as in F-N, M-L or H-H
• Method of Lines
• Stiffness issue stiffness of the system

35
Modeling Methods Analysis

• Computation
• Numerics
• MATLAB, Mathematica, XTC
• Individual code in MATLAB Mathematica
• Available on-line via webMathematica
• XTC recommended
• XPPAUT
• sample code
• animation
• Simulations

36
Modeling Methods Analysis

• Computation
• Numerics
• MATLAB, Mathematica, XTC
• Available on-line via webMathematica
• http//math.jccc.net8180/webMathematica/MSP/mmart
  in/morlecar

37
webMathematica output
38
XPPAUT Code

• collapse5.ode more oscillators
• lewis parameters plus diffusion
• increased d to get larger wave, increased k for
  sing limit, decreased d2
• param a.1,d3,d2.025,d3.025,eps.05,gamma0,i0
  .08
• param p1-0.01,p20.15,p30.1,p40.145
• param gl0.5,gk2.0,gca1.0,eca1.0,ek-0.7,el-0.
  5,k100.0
• i v00.2,v10.2,v20.2,v30.2,v40.21,v50.22,v60
  .23,v70.24,v80.25,v90.26
• i v100.27,v110.28,v120.29,v130.3,v140.31,v15
  0.32,v160.33,v170.34
• i v180.35,v190.36,v200.37,v210.38,v220.39,v23
  0.4,v240.41,v250.42
• i v260.43,v270.44,v280.45,v290.46,v300.47,v31
  0.48,v320.49,v330.5,v340.5
• i v350.2,v360.2,v37-.1,v38-.1
• i v39..160-.4
• i w00.3,w10.3,w20.3,w30.3,w40.29,w50.28,w60
  .27,w70.26,w80.25,w90.24
• i w100.23,w110.22,w120.21,w130.2,w140.19,w15
  0.18,w160.17,w170.16,w180.15
• i w190.14,w200.13,w210.12,w220.11,w230.1,w24
  0.09,w250.08,w260.07
• i w270.06,w280.05,w290.04,w300.03,w310.02,w32
  0.01,w330,w340,w350,w360,w370
• w38..160(0)0.1
• winft(v)0.5(1tanh((v-p3)/p4))
• minft(v)0.5(1tanh((v-p1)/p2))

wave.ani animated wave fcircle
.05.0060..150.5(vj1).011-wj end
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Outline of Fitzhugh-Nagumo Model
Type
Diagram
Parameters
Phenomena
A.
a.12134 b.121372227 c.12490008
a.1256010 b.1256013959397246 c.13084809567353
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B.
a.1339531138011688 b.13402998 c.14690736 d8
.87047
C.
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Modeling Methods Analysis

• Computation
• Fitzhugh-Nagumo
• Case A, 1
• Par 0 lt d2 0.12134
• Result Block

41
Modeling Methods Analysis

• Computation
• Fitzhugh-Nagumo
• Case A, 2
• Par1 0.12134 d2 0.12137
• Par2 0. 12137 lt d2 0.121372227
• Result Propagation

42
Modeling Methods Analysis

• Computation
• Fitzhugh-Nagumo
• Case A, 3
• Par 0.121372227 lt d2 0.12490008
• Result Reflection

43
Modeling Methods Analysis

• Computation
• Fitzhugh-Nagumo
• Case A, 4
• Par1 0.12490008 lt d2 0.126
• Par2 0.126 lt d2
• Result Propagation

44
Modeling Methods Analysis

• Computation
• Fitzhugh-Nagumo
• Case B, 1
• Par 0 lt d2 0.1256010
• Result Block

45
Modeling Methods Analysis

• Computation
• Fitzhugh-Nagumo
• Case B, 2
• Par 0.1256010 lt d2 0.1256013959397246
• Result Propagation

46
Modeling Methods Analysis

• Computation
• Fitzhugh-Nagumo
• Case B, 3
• Par 0.1256013959397246 lt d2
  0.1308480956735347
• Result Reflection

47
Modeling Methods Analysis

• Computation
• Fitzhugh-Nagumo
• Case B, 4
• Par1 0 .1308480956735347 lt d2 0.139
• Par2 0.139lt d2
• Result Propagation

48
Modeling Methods Analysis

• Computation
• Fitzhugh-Nagumo
• Case C, 1
• Par 0 lt d2 0.1339531138011688
• Result Block

49
Modeling Methods Analysis

• Computation
• Fitzhugh-Nagumo
• Case C, 2
• Par1 0.1339531138011688 lt d2 0.13402997
• Par2 0. 13402997 lt d2 0.13402998
• Result Propagation

50
Modeling Methods Analysis

• Computation
• Fitzhugh-Nagumo
• Case C, 3
• 0.13402998 lt d2 0.14690736
• Result Reflection

51
Modeling Methods Analysis

• Computation
• Fitzhugh-Nagumo
• Case C, 4
• Par1 0.14690736 lt d2 0.1474
• Par2 0.1474 lt d2 8.867655
• Par3 8.867655 lt d2 8.87047
• Result Propagation

52
Modeling Methods Analysis

• Computation
• Fitzhugh-Nagumo
• Case C, 5
• Par 8.87047 lt d2
• Result Block

53
Modeling Methods Analysis

• Computation
• Morris-Lecar
• Case A
• d 0.01
• Blocking

54
Modeling Methods Analysis

• Computation
• Morris-Lecar
• Case A
• d 0.015
• 0.0141 lt d lt 0.0178
• Reflection transmission

55
Modeling Methods Analysis

• Computation
• Morris-Lecar
• Case A
• d 0.04
• Transmission

56
Modeling Methods Analysis

• Computation
• Morris-Lecar
• Case B
• d 0.01
• 0.0141 lt d lt 0.0178
• Blocking

57
Modeling Methods Analysis

• Computation
• Morris-Lecar
• Case B
• d 0.015
• 0.0141lt d lt0.0178
• Reflection transmission

58
Modeling Methods Analysis

• Computation
• Morris-Lecar
• Case B
• d 0.04
• 0.0141 lt d lt 0.0179
• Transmission

59
Modeling Methods Analysis

• Computation
• Morris-Lecar
• Case C
• d2 0.04
• d3 0.01
• Blocking

60
Modeling Methods Analysis

• Computation
• Morris-Lecar
• Case C
• d2 0.025
• d3 0.015
• Reflection transmission

61
Modeling Methods Analysis

• Computation
• Morris-Lecar
• Case C
• d2 0.04
• d3 0.015
• Reflection transmission, both

62
Modeling Methods Analysis

• Computation
• Morris-Lecar
• Case C
• d2 0.025
• d3 0.025
• Reflection transmission, both

63
Modeling Methods Analysis

• Computation
• Morris-Lecar
• Case C
• d2 0.015
• d3 0.025
• Reflection transmission, both

64
Biological Interpretation

• Conduction of action potential
• Threshold exceeded
• Ion channel activity initiated
• Adjacent portions of membrane stimulated
• Traveling wave generated
• Effect of abrupt increase in diameter of axon
• Decrease in membrane current density ahead of
  traveling wave
• Greater amount of membrane to be charged
• Decrease in speed and amplitude of wave
• Delay

65
Biological Interpretation

• Conduction Block
• Abrupt increase in axon diameter sufficiently
  large
• Insufficient current to charge the larger amount
  of membrane
• Echo Waves
• Abrupt increase in axon diameter
• Delay of action potential
• Encounter with refractory period for portion of
  membrane
• Sufficiently long to initiate depolarization of a
  repolarized part

66
Summary and Conclusions

• Numerical Experiments
• Bistable equation
• FitzHugh-Nagumo
• Morris-Lecar
• Phenomena Observed
• Blocking and Conduction
• Echo
• Critical values for each case

67
Summary and Conclusions

• Parameter Sensitivty Echo
• Y. Zhao, J.Bell, Mathematical Biosciences, 1994
• FitzHugh-Nagumo
• R. Altenberger, et. al., J. Neuroscience Methods,
  2001
• Morris-Lecar
• Hodgkin-Huxley
• Lack of traveling wave solution

68
Questions and Future Problems

• Different Geometries
• Taper Axon Hillock
• Echo trap
• Hodgkin-Huxley
• Careful numerical studies
• Echo
• Mechanism for Pain

69
References (1 of 2)

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• Fall, C.P., et al Computational Cell Biology,
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• Hasting, S.P. On the Existence of Homoclinic and
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• C.K.R.T Jones, Stability of the traveling wave
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References (2 of 2)

• J.P. Keener, J. Sneyd. Mathematical
  Physiology, Springer-Verlag, 1998
• T.J. Lewis, J.P. Keener, Wave-block in excitable
  media due to regions of depressed excitability,
  SIAM J. Appl. Math., 611293316, 2000
• Miller, C.C.J., et al, Axonal transport of
  neurofilaments in normal and disease states,
  Cell. Mol. Life Sci., 59323-330, 2002
• Murray, J.D. Mathematical Biology Vol. I,
  Springer-Verlag, 2003
• J. P. Pauwelussen, Nerve impulse propagation in
  a branching nerve system A simple model,
  Physica 4D, 6788, 1981.
• J. P. Pauwelussen, One way traffic of pulses in
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• F. Ramón, R.W. Joyner, J. W. Moore, Propagation
  of action potentials in inhomogeneous axon
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  Proceedings, 34513571363, 1975.
• Sajda, P. Computational Neural Modeling and
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• A. Scott, Neuroscience A Mathematical Primer,
  Springer-Verlag, New York, 2002.
• D. Terman, An introduction to dynamical systems
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• L. Wang, A. Brown, Rapid Intermittent movement
  of axonal neurofillaments observed by
  fluorescence photobleaching, Molecular Biology
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71
MBI Project Team
From left to right Hsiu-Tsun Hsieh, Prasanna
Karunanayaka, Yevgeniy Gokun, Namyong Lee, Avner
Friedman, Mike Martin, Stephen Clark
72
Additional Pointers Continuing Resources

• Mike Martin
• mmartin_at_jccc.net
• http//www.jccc.net/mmartin/mbi/

mbi.osu.edu