Noiseinduced motion in neural network and its control

1
Noise-induced motion in neural network and its
control

• Natalia Janson
• with
• Sandhya Patidar, Andrey Pototsky

2
Model of a Single Excitable Neuron

• FitzHugh-Nagumo system

Timescale separation parameter Distance from the
bifurcation point Gaussian white noise Noise
strength (temperature)
Parameters
t
3
Model of a NETWORK of Excitable Neurons

• N FitzHugh-Nagumo oscillators with independent
  Gaussian white noise sources ?i , coupled through
  the mean field Mx

Mean field
Parameters
4
Possible Behavior of the Network Mean Field
Oscillations around the Fixed Point
Dynamical chaos
From Zaks, Sailer, Schimansky-Geier, Neiman,
Chaos 15, 026117 (2005)
5
Possible Behavior of the Network Mean Field
Spiking
Chaotic spiking (intermittent)
Temperature grows
Larger temperature!
Zaks, Sailer, Schimansky-Geier, Neiman, Chaos
15, 026117 (2005)
6
Possible Behavior of the Network Mean Field
Spiking
Chaotic spiking (intermittent)
Smaller T
Larger T
Periodic oscillations
7
Map of Regimes on the Parameter Plane? and T
No spiking outside l1
No oscillations outside l2
From Zaks, Sailer, Schimansky-Geier, Neiman,
Chaos 15, 026117 (2005)
8
Transition to Synchronization
Variance of mean field vs coupling ?

• ?

?
?
9
Role of Stochastic Synchronization in Neural
Network

• Constructive
• Enhances the processing of biological
  information between neurons.
• 2. Destructive
• Can induce a regular, rhythmic activity, which
  is believed to play a crucial role in the
  emergence of diseases like Parkinsons Disease,
  Epilepsy and Essential Tremor.

10
A Good Control Tool Can

• Enhance or suppress stochastic synchronization
• Manipulate timescales of stochastic oscillations
• Manipulate the regularity of oscillations

• and Cannot
• Do any harm should be almost non-invasive

11
Method of Control TDFC

• Time-Delayed Feedback Control (TDFC),
• originally developed by K. Pyragas
• Pyragas 1992 for deterministic chaos.

Control Force Delayed feedback P (t) K(x(t -
t ) - x(t))
12
Delayed Feedback Control of Chaos What to Expect?

• Chaos can turn into periodic orbit
• ANY oscillations can be stopped
• Timescale can be changed
• Multistability and hysteresis can occur!

13
TDFC of Chaos Bifurcation Analysis

• Known bifurcations on the plane (? - K) for
• DETERMINISTICALLY CHAOTIC
• SYSTEMS

Rossler system in CHAOTIC regime Balanov,
Janson, Scholl, Phys. Rev. E 2005
14
Delayed Feedback Control of Chaos What to
Expect?
15
Delayed Feedback Control for Noise-Induced
Oscillations?

• Control of noise-induced oscillations in a single
  excitable unit
• Manipulation of properties of stochastic limit
  cycle.
• Janson, Balanov, Schöll, Phys. Rev. Lett. 2004.
• Manipulation the dynamics of two coupled
  excitable units.
• Hauschildt, Janson, Balanov, Schöll, Phys. Rev.
  E 2006

16
Delayed Feedback Control of Noise-Induced
Oscillations in a Single FitzHugh-Nagumo Unit
Degree of order can be changed
Time scales of oscillations can be changed
17
Network of Excitable Units with TDFC

• N FitzHugh-Nagumo oscillators
• with independent Gaussian white noise sources ?i
• coupled through mean field Mx
• with control applied through My

Feedback Parameters
18
What can TDFC do Induce spiking

• Without TDFC Sub-threshold oscillations ?
  0.01, a 1.05, ? 0.1

No control
With control
K 0.1, ? 1.4
K 1.0, ? 0.8
19
What can TDFC do Regularize or Suppress Spiking

• Without TDFC Chaotic spiking ? 0.01, a
  1.05, ? 0.1.

No control
With control
K 1.0, ? 0.8
K 0.1, ? 0.73
20
A Bit of Analytics

• Model N nonlinear stochastic differential
  equations.
• Major problems for analytical treatment
• N is a large number
• Equations are stochastic
• Solution Describing only the statistical
  properties of the model.
• How? By obtaining a system of equations for
  MOMENTS of
• distribution.
• Are they good? Yes Moment (cumulant) equations
  are deterministic.
• Anything bad about them? Yes there is an
  infinite number of
• them.

21
Probability Density DistributionNo Coupling -
No Synchronization
22
Probability Density DistributionCoupling --
Synchronization!
23
Gaussian Approximation Method

• If the distribution of variables in the ensemble
  were Gaussian, there would be only 5 non-zero
  moments in our network.
• They are 2 means, 2 variances, 1 cross-variance.
  
• Make a Gaussian approximation
• Assume the distribution of system variables is
  approximately Gaussian.
• Then neglect the moments of higher orders.
• We are left with
• 5 deterministic equations for statistical
  averages!

• Without TDFC Zaks, Sailer, Schimansky-Geier,
  Neiman, Chaos 15, 026117 (2005)
• Method used
• Rodriguez and Tuckwell, Phys. Rev. E 54, 5585
  (1996)
• Tanabe and Pakdaman, Phys. Rev. E 63, 031911
  (2001)

24
Cumulant Equations with TDFC

• Mean fields Mx ?xi?, My ?yi?,
• Variances Dx ?(xi - Mx)2?, Dy ?(yi -
  My)2?,
• Covariance Dxy ?(xi - Mx) (yi - My)?

25
TDFC Bifurcation Analysis of Cumulant Equations.
Subthreshold Chaos
Fixed point
No control
Spiking (chaotic or periodic)
26
TDFC Bifurcation Analysis of Cumulant Equations.
Subthreshold Chaos
No control
With control
27
TDFC Bifurcation Analysis of Cumulant Equations.
Subthreshold Chaos
Periodic spiking
Chaotic spiking
28
Conclusions

• In a network of excitable units time-delayed
  feedback can
• induce synchronization
• enhance synchronization
• suppress synchronization
• change the coherence of oscillations
• change the timescale of oscillations (not very
  well, however)

29
Are cumulant equations adequate?

• Gaussian approximation is not very accurate, and
  the cumulant equations are not in a quantitative
  agreement with Langevin equations.
• However qualitative agreement is in place.
• Work in Progress
• Considering moments of higher orders
• Does it help? YES.

30
Acknowledgements

• Alexander Balanov, for helpful discussions
• EPSRC, UK, for funding

31
Synchronization From Simple to Complex
New book by Springer coming
for students and scientists
Authors about the Book

• Table of Contents
• 1. Introduction
• Part I Mechanisms of Synchronization
• 2. General Remarks
• 3. 11 Forced Synchronization of Periodic
• Oscillations
• 4. 11 Mutual Synchronization of Periodic
• Oscillations
• 5. Homoclinic Mechanism of Synchronization
• of Periodic Oscillations
• 6. nm Synchronization of Periodic Oscillations
• 7. 11 Forced Synchronization of Periodic
  Oscillations in the Presence of Noise
• 8. Chaos Synchronization
• 9. Synchronization of Noise-Induced Oscillations
• 10. Conclusions to Part I

Alexander Balanov, Nottingham University, United
Kingdom Interaction of systems with irregular
(chaotic and stochastic) dynamics is a problem
that arises in a number of branches of modern
science. It has been known for quite a while now,
that in spite of their complexity, such
oscillations can be synchronized. However, it is
really amazing that the physical mechanisms of
synchronization of unpredictable chaotic and even
stochastic oscillations are identical to the ones
inherent in the quite predictable periodic
motion. The universality of synchronization is an
important concept in understanding of the
functioning of complex systems that contain
coupled oscillators". Natalia Janson,
Loughborough University, United Kingdom
Synchronization plays the central role in
modern science of oscillations and waves.
However, classical results that form the basis of
synchronization concept are not easily
accessible, in particular for young scientists
and students they are distributed over a huge
number of publications in different languages,
and often described in a style which
non-mathematicians consistently find difficult to
follow. Also, synchronization is a real-life
phenomenon that manifests itself in adjustment of
measurable physical quantities like timescales
and amplitudes. This books establishes the link
between its mathematical and physical
descriptions, while presenting the material at a
level suitable for interested people with only
basic mathematical background." Dmitry
Postnov, Saratov State University,
Russia Turning over the pages of our
scientific papers I realized that they are anyhow
related to different aspects of synchronization
in various systems. I like the idea of combining
in one book the classical approach and up-to-date
understanding of synchronization of complex
systems". Olga Sosnovtseva, Technical
University of Denmark I believe that the
concept of synchronization can be applied to
improve our understanding of complex systems of
interacting biological oscillators. Although many
cells in the body display intrinsic spontaneous
rhythmic activity, physiological function derives
from the interactions of these cells with each
other and with external environment in order to
generate rhythms that are essential for life.
Synchronization refers to the way in which the
networked elements communicate and exhibit
collective behaviour. "

• Book Features
• Key classical results are derived in every
  detail and explained from different angles no
  steps of calculations are omitted, no it is easy
  to show or it is obvious remarks.
• Most synchronization mechanisms are illustrated
  with full-scale electronic or biological
  experiments.

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