Modeling the complexity of pain – possibilities and problems

Dorsal horn

Nevropeptidergisk system

Opioidergisk system

Dorsal horn 2006 (?)

Transduction (neurotrophins)

Other sources for complexity

• Extrasynaptic effects of transmitter substances

• Functional effects of astrocytes

?

Santa Fe institute

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.

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.

Approaches

• The Bottom-Up approach

• The hidden signals approach

• The Top-down approach

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 that a wealth of potential valuable information that may affect clinical care remains largely an untapped resource” Goldstein 2003

• Signals with fractal properties, and with different magnitude of entropy can be analyzed

Recent example

• 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

Approximate Entropy• Natural information parameter for an approximating

Markov Chain; closely related to Kolmogorov entropy• Can be applied to short, noisy series

– 100 < N < 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; 266:H1643

• 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; 278:H2039.

• 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

ApEN time course of WDR neurons receiving peripheral nociseptive input

Firing rate of WDR neurons receiving nociseptive input: spike number does not

correlate with ApEN

Firing rate of WDR neurons after morphine injection: Correlation with ApEN

The bottom-up approach

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

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 its slow potential

Schematic diagram of the model

Equations

• The slow potential is defined as:

V(t) = 1/s t

t-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)

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)

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

Equation IV

• Differentiating (3) using (1) and (2) yields:

(4) kVk = - (Vk – Vk0) + jkgjk(xj)

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)

Results I: C fiber stimulation – increase of nociseptive output from the dorsal horn

Results II: Large fiber activation leads to a transient increase, later to a decrease of nociseptive signals from the dorsal horn

Results III: small fiber stimulation with Hz leads to wind-up like phenomena

Top-down approach

Coupling K12

K1

K2

Equations

),,()()()()( 00202 ptftR iiii

N

jijjiiijiiii

),,()()()1( 222 ptftR ii

N

jijiijiiiiii

),),(()())()(())(1)(()1( pnnfnRnnknnmn ii

N

jijiijiii

Coupled linear (harmonic) stochastic oscillators

Coupled nonlinear (e.g. Van-der-Pol) stochastic oscillators:

Coupled nonlinear stochastic iterative maps (e.g. logistic map):

i - characteristic rate of the organ (e.g. beat-to-beat interval)

i i - oscillators frequency and damping coefficients

),,( ptf

- “treatment function” (the influence of a medical device, e.g. mechanical ventilator)

The time series were analyzed using the following methods:

• Power Spectra (Power Law);• Standard Deviation (STD);• Approximate Entropy (ApEn).

Approximate entropy and standard deviation as functions of the coupling coefficient for the case of two (left) and five (right) “organ” systems.

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.

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”.

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).

Conclusions

Powersim models

• Powersim is a graphical system for quantitative dynamical systems

• Similar programs include Stella and Vensim

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

f(x) = 100 * exp ((x-0.4)/12) 100 + exp ((x-0.4)/12)

01 02 030

20

40

60

80

100

subjective pain

Nociseption

Nur für den nichtkommerziellen Einsatz!

Nociseption

Nociseptive Input Antinociseption

subjective pain

Nociseption

Nociseptive Input

subjective pain

Random variable

Normal distribution

Antinociseption

Opioids

Submodel for PCA-profile

Timer

StartTimer ReduceTimer

Profile1

Profile2

subjective pain

Profile3

Profile4

Submodel for pharmacokinetics

TimerOpioids

Analgetic Dose 1

Analgetic Dose 2

Analgetic Dose 3

Analgetic Dose 4

Pharmakinetics

Hilfsvariable_ 1

Hilfsvariable_ 2

Hilfsvariable_ 3

Hilfsvariable_ 4

Whole „PCA-model“

Nociseption

Nociseptive Input

subjective pain

Random variable

Normal distribution

Antinociseption

Timer

StartTimer ReduceTimer

Profile1

Profile2

subjective pain

Profile3

Profile4

TimerOpioids

Analgetic Dose 1

Analgetic Dose 2

Analgetic Dose 3

Analgetic Dose 4

Pharmakinetics

Hilfsvariable_ 1

Hilfsvariable_ 2

Hilfsvariable_ 3

Hilfsvariable_ 4

0,12

-24,62

Typical result

01 02 030

20

40

60

80

100

subjective pain

Opioids (1/ hr)

Nur für den nichtkommerziellen Einsatz!

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