Complexity and pain

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Complexity and pain
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• The problem
• Definitions and explanations of complex systems,
  chaos, dynamics and emergence
• Approaches in pain (and palliative care) with
  examples

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Complexity of pain on the molecular level
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Dorsal horn
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Neuropeptid system
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Opioid system
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Cholezystokinin
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Transduction (e.g. neurotrophins)
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Polysynaptic modulation

• Transmitter substances have also effects outside
  of the synaptic cleft
• Increased transmitter concentrations can lead to
  a rather excitatoric or inhibitoric tone
• This regards specially G-protein coupled receptors

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Glia as neuromodulator

• Astrocytes and glia eksprime receptors similar to
  neurons (evidence beyond others for NMDA, NK1,
  P2X4)
• They send out transmitter substances (bl.a. SP,
  CGRP, glutamat), cytokines (bl.a. Il1, Il6, TNF)
  and modulate elimination of neurotransmitters
• Possibly an important function in neuropathic
  pain Tsuda 2003

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Watkins 2002
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?
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Santa Fe institute
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Principles of Complex Systems

• Parts distinct from system.
• System displays emergent order.
• Not directly related to parts.
• Define nature function of system.
• Disappear when whole is broken up.
• Robust stability (basins of attraction).
• Emergent order systemic properties which define
  health vs. disease.

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Principles of Complex Systems

• Unpredictable response
• Chaos or sensitivity to initial conditions.
• Response determines impact to system
• Controlled experiments reproducible results
• Uncontrolled patients unpredictable results
• Host response is unpredictable,
• yet it determines outcome.

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Some explanations

• Chaos (with help of Larry Liebovitch, Florida
  Atlantic University)
• Dynamics (with help of J.C.Sprott, Department of
  Physics,University of Wisconsin Madison)

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These two sets of data have the same

• mean
• variance
• power spectrum

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RANDOM random x(n) RND
Data 1
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CHAOS Deterministic x(n1) 3.95 x(n) 1-x(n)
Data 2
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etc.
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RANDOM random x(n) RND
Data 1
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CHAOS deterministic x(n1) 3.95 x(n) 1-x(n)
Data 2
x(n1)
x(n)
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CHAOS
Definition
predict that value
Deterministic
these values
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CHAOS
Definition
Small Number of Variables
x(n1) f(x(n), x(n-1), x(n-2))
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CHAOS
Definition
Complex Output
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CHAOS
Properties
Phase Space is Low Dimensional
d , random
d 1, chaos
phase space
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CHAOS
Properties
Sensitivity to Initial Conditions
nearly identical initial values
very different final values
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CHAOS
Properties
Bifurcations
small change in a parameter
one pattern
another pattern
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Time Series
X(t)
Y(t)
Z(t)
embedding
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Phase Space
Z(t)
phase space set
Y(t)
X(t)
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Phase Space
Constructed by direct measurement
Measure X(t), Y(t), Z(t)
Z(t)
Each point in the phase space set has
coordinates X(t), Y(t), Z(t)
X(t)
Y(t)
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Analyzing Experimental Data
The Good News
In principle, you can tell if the data was
generated by a random or a deterministic
mechanism.
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Analyzing Experimental Data
The Bad News
In practice, it isnt easy.
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Dynamics the predator-prey example

• A dynamic system is a set of functions (rules,
  equations) that specify how variables change over
  time.

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Rabbit Dynamics

• Let R of rabbits
• dR/dt bR - dR

rR
r b - d
Birth rate
Death rate

• r gt 0 growth
• r 0 equilibrium
• r lt 0 extinction

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Exponential Growth

• dR/dt rR
• Solution R R0ert

R
r gt 0
r 0
rabbits
r lt 0
t
time
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Logistic Differential Equation

• dR/dt rR(1 - R)

1
R
r gt 0
rabbits
t
0
time
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Effect of Predators

• Let F of foxes
• dR/dt rR(1 - R - aF)

Intraspecies competition
Interspecies competition
But The foxes have their own dynamics...
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Lotka-Volterra Equations

• R rabbits, F foxes
• dR/dt r1R(1 - R - a1F)
• dF/dt r2F(1 - F - a2R)

r and a can be or -
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Types of Interactions
dR/dt r1R(1 - R - a1F) dF/dt r2F(1 - F - a2R)

a2r2
Prey- Predator
Competition
-

a1r1
Predator- Prey
Cooperation
-
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Equilibrium Solutions

• dR/dt r1R(1 - R - a1F) 0
• dF/dt r2F(1 - F - a2R) 0

Equilibria

• R 0, F 0
• R 0, F 1
• R 1, F 0
• R (1 - a1) / (1 - a1a2), F (1 - a2) / (1 -
  a1a2)

F
R
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Stable Focus(Predator-Prey)
r1(1 - a1) lt -r2(1 - a2)
r1 1 r2 -1 a1 2 a2 1.9
r1 1 r2 -1 a1 2 a2 2.1
F
F
R
R
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Stable Saddle-Node(Competition)
a1 lt 1, a2 lt 1
r1 1 r2 1 a1 1.1 a2 1.1
r1 1 r2 1 a1 .9 a2 .9
Node
Saddle point
F
F
Principle of Competitive Exclusion
R
R
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Coexistence

• With N species, there are 2N equilibria, only one
  of which represents coexistence.
• Coexistence is unlikely unless the species
  compete only weakly with one another.
• Diversity in nature may result from having so
  many species from which to choose.
• There may be coexisting niches into which
  organisms evolve.
• Species may segregate spatially.
• Purely deterministic
• (no randomness)
• Purely endogenous
• (no external effects)
• Purely homogeneous
• (every cell is equivalent)
• Purely egalitarian
• (all species obey same equation)
• Continuous time

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Typical Results
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Typical Results
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Approaches

• The Bottom-Up approach
• The hidden signals approach
• The Top-down approach

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Hidden signals approach

• Neurons and other structures do not only
  communicate by action potential frequencies
• Despite the multitude of physiologic signals
  available for monitoring, we suggest tha a wealth
  of potential valuable information that may affect
  clinical care remains largely an untapped
  resource Goldstein 2003

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• Dopaminergic neurons in Striatum (N. accumbens)
  can send two different informations, one
  frequency coded and one pattern (burst activity)
  coded Fiorillo 2003
• Burst patterns can transport informations over
  stimulation properties Krahe 2004

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Approximate Entropy

• Natural information parameter for an
  approximating Markov Chain closely related to
  Kolmogorov entropy
• Can be applied to short, noisy series
• 100 lt N lt 5000 data points
• Conditional probability that two sequences of m
  points are similar within a tolerance r
• Pincus SM, Goldberger AL, Am J Physiol 1994
  266H1643
• Other entropy measures
• Cross-ApEn - compare two related time series
• Sample entropy - does not count self matches
• Richman JS, Moorman JR, Am J Physiol Heart Circ
  Phys 2000 278H2039.

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• Inflammatory pain model
• Identification of WDR Neurons in L4/L5 with their
  receptive fields
• Injection of bee venom
• WDR neurons showed different stable ApEN values

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ApEN time course of WDR neurons receiving
peripheral nociseptive input
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Firing rate of WDR neurons receiving nociseptive
input spike number does not correlate with ApEN
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Firing rate of WDR neurons after morphine
injection Correlation with ApEN
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Heart rate variability
Publications about HRV
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HRV-analyse which of the four time-domain
analysis are pathological ?
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Goldberger 2002 A,C kongestive heart failure,D
atrial fibrillation
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HRV changes in palliative patients

• Inclusion with a life expectancy less than 6
  months
• 10 minutes ECG (Biocom) with HRV variables (SDNN,
  frequency domain, ApEN)
• ECG recordings every time patient is in or comes
  to clinic, aimed weekly (but no extra
  consultations for study purposes)
• 24 patients to be included, until now 4 patients

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• 55 years old, male, healthy
• 1996 radiation therapy of esephagus because of
  atypical cells
• 7/2005 diagnosis of esephageal cancer with
  metastases.
• 8/2005 stent
• 8/2005 cytostatics (ELF)
• 9/2005 Stroke because of brain metastasis
• 10/2005 Study inclusion. Increasing CRP, but
  normal PCT
• 11/2005 stop of cytostatical therapy because of
  tumor progression
• 2/12 Big amounts of ascites
• 8.1.2006 Pat deceased

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The bottom-up approach

• Models to simulate physiologic and
  pathophysiologic dynamics
• Mathematical models
• Network models
• Graphical models
• Agent based systems

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Britton/ Chaplain/ Skevington (1996)The role of
N-methyl-D-aspartate (NMDA) receptors in wind-up
a mathematical model

• Base assumptions in the model
• One C-fiber, one A?-fiber, which are connected to
  a transmission cell (WDR) in the dorsal horn.
• The transmission cells gets input from inhibitory
  and excitatory interneurons and is sending
  signals to cells in the midbrain
• Midbrain cells again send inhibitory signals
  directly to transmission cells and excitatory
  signals to inhibitory interneurons
• The firing frequency of a particular cell is a
  function of ist slow potential

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Schematic diagram of the model
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Equations

• The slow potential is defined as
• V(t) 1/s ?tt-1 V (?) d ?
• ( s interval of time)
• Frequencies xi, xe, xt, xm are functions of the
  slow potentials
• xi ?i(Vi), xe (Ve), xt ?t(Vt), xm ?m(Vm)

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Equations II

• The effekt of an input frequency xj to a synapse
  of a cell of potential Vk will be to raise it by
  ?jk
• (1) ?jk ?jk ?t-? hjk (t-?) gjk (xj (?)) d?
• Where ?jk is equal to 1 for an excitatory synapse
  and -1 for an inhibitory synapse, hjk is a
  positive montone decreasing function and gjk is a
  bounded, strictly monotone increasing function
  satisfying gjk (0) 0
• The simplest form for hjk is
• (2) hjk ( t) 1/ ?k exp (- 1/ ?k)

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Equations III

• The total effect of all inputs on the potential
  of cell k gives
• (3) Vk Vk0 ? ?jk
• Assuming that the system is linear

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Equation IV

• Differentiating (3) using (1) and (2) yields
• (4) ?kVk - (Vk Vk0) ??jkgjk(xj)

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Model equations

• (5) ?iVi - (Vi-Vi0) gli(xI)gmi(xm)
• (6) ?eVe -(Ve-Ve0) gse (xs,Ve)
• (7) ?tVt -(Vt-Vt0) gst(xs) glt(xl)
  get(xe) git(xi) gmt (xm)
• (8) ?mVm -(Vm-Vm0) gtm(xt)

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Results I C fiber stimulation increase of
nociseptive output from the dorsal horn
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Results II Large fiber activation leads to a
transient increase, later to a decrease of
nociseptive signals from the dorsal horn
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Results III small fiber stimulation with Hz
leads to wind-up like phenomena
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Top-down approach
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Possible Role of Coupling between the Organs in
the Development of Systemic Illnesses Model of
Coupled Stochastic Oscillators Anton Burykin,
Gernot Ernst, Andrew JE Seely Department
of Chemistry, University of Southern California,
Los Angeles (bourykin_at_usc.edu, http//futura.usc.e
du/wgroup/people/anton/index.htm) Kongsberg
Hospital, Norway (gernot.ernst_at_blefjellsykehus.no)
Divisions of Thoracic Surgery Critical
Care Medicine, University of Ottawa, Ottawa,
Ontario, Canada (aseely_at_ottawahospital.on.ca)
The Approach
Preliminary Results Stochastic Harmonic
Oscillators
Motivation
The origin of the statistical differences between
the patterns of variation of physiological
signals (e. g. heart rate or respiratory
variability) in health compared to illness states
or aging remains a clinically relevant and
unsolved problem 1, 2. A current hypothesis 2
suggests that states involving systemic illness,
e.g. multiple organ dysfunction (MOD) or aging,
occur due to the uncoupling and/or change in
communication between single organs or
subsystems. In order to test this hypothesis, we
suggest here several possible dynamical models
(constructed from the network of coupled
elements) of the multiple organ system (the
organism) and compare their behavior in different
regimes of coupling between their elements.
A. Effect of Uncoupling
B. Constant Frequency vs. Noisy Mechanical
Ventilation
Models

• organism is represented as a system of
  organs
• each organ has its own rhythm (rate or
  frequency), which itself is changing with time
• organs (and their rhythms) are coupled to
  each other (the physiological examples of
  coupling include neural, humoral, mechanical,
  etc).

log(P)
Fig 1. Approximate entropy and standard deviation
as functions of the coupling coefficient for the
case of two (left) and five (right) organ
systems.
log(f)
Fig 2. Simulation of the effect of mechanical
ventilation with the constant frequency for the
case of the organism with just two organs.
(left) Approximate entropy and standard deviation
of the first organ in the control and during
the fixing the frequency of the second one for
several values of the coupling between the
organs. (right) Change of the slope in log-log
representation of the power spectra of the first
oscillator.
Coupled linear (harmonic) stochastic oscillators
Conclusion
These results supports the hypothesis that
decreased coupling or fixing the frequency of a
single organ leads to decreased complexity of a
system. A clinical example of decreased coupling
or maintaining a fixed rate of organ oscillation
includes certain modes of mechanical ventilation.
These mechanisms may represent a mechanism by
which clinical deterioration may lead to
progressive refractory organ dysfunction. Future
investigations regarding the model would include
an evaluation of the current hypothesis using
non-linear, time-delayed or time-dependent
coupling and use of different treatment
functions (medical devises).
Coupled nonlinear (e.g. Van-der-Pol) stochastic
oscillators
Acknowledgments
Coupled nonlinear stochastic iterative maps (e.g.
logistic map)
This work was supported by the grant GM62674
from the National Institutes of Health to the
Santa Fe Institute. We are grateful to Timothy G.
Buchman and Lee Segel for useful discussions.
- characteristic rate of the organ (e.g.
beat-to-beat interval)
- oscillators frequency and damping coefficients
References
Ri(t) random force
- treatment function (the influence of a
medical device, e.g. mechanical ventilator)
(1). A. L. Goldberger, L. A. N. Amaral, J. M.
Hausdorff, P. Ch. Ivanov, C.-K. Peng, and H. E.
Stanley, Fractal Dynamics in Physiology
Alterations with Disease and Aging, Proc. Natl.
Acad. Sci. 99, 2466-2472 (2002). (2). Godin PJ,
Buchman TG. Uncoupling of biological oscillators
a complementary hypothesis concerning the
pathogenesis of multiple organ dysfunction
syndrome. Crit Care Med. 1996 Jul24(7)1107-16. (
3). Pincus, S. M. Approximate entropy as a
measure of system complexity. Proc Natl Acad Sci
USA 1991882297-2301. (4). Seely AJ, Macklem PT.
Complex systems and the technology of variability
analysis. Crit Care. 2004 Dec8(6)R367-84. Epub
2004 Sep 22. (5). Pincus, S. M. Greater signal
regularity may indicate increased system
isolation. Math Biosci. 1994 Aug 122(2)161-81.
Fig. 3. Simulation of the effect of mechanical
ventilation with the constant frequency for the
case of the organism with four organs. (left)
control, (right) effect of the fixing the
frequency of the 3rd organ.

• The time series were analyzed using the following
  methods
• Power Spectra (Power Law)
• Standard Deviation (STD)
• Approximate Entropy (ApEn).

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Powersim models

• Powersim is a graphical system for quantitative
  dynamical systems
• Similar programs include Stella and Vensim

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General assumptions for the model

• Nociseption and antinociseption are tonically
  active and in balance
• Nociseption can be modeled quantitatively
• Subjective pain and emotions can be modeled as
  VAS between 1 and 100
• To redimension VAS scales to an area from 0 to
  100, a logistic differential equation was used

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f(x) 100 exp ((x-0.4)/12)
100 exp ((x-0.4)/12)
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Submodel for PCA-profile
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Submodel for pharmacokinetics
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Whole PCA-model
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Typical result
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Summary

• Nociseptive systems are complexe systems and
  share common properties with other complexe
  (adaptive) systems
• There exist three main approaches to get insight
  in complex systems Bottom up, Hidden signals,
  Top down.
• While being succseful used in other areas
  (Neurobiology, Cardiology), the contribution of
  complex systems theory in pain research sounds
  promising, but is yet unclear

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