On the Dynamics of Delayed Neural Feedback Loops. Sebastian Brandt Department of Physics, Washington University in St. Louis

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1 On the Dynamics of Delayed Neural Feedback Loops Sebastian Brandt Department of Physics, Washington University in St. Louis

2 Overview of Dissertation Chapter 2: S. F. Brandt, A. Pelster, and R. Wessel, Variational calculation of the limit cycle and its frequency in a two-neuron model with delay, Phys. Rev. E 74, /1-14 (2006). Chapter 3: S. F. Brandt, A. Pelster, and R. Wessel, Synchronization in a neuronal feedback loop through asymmetric temporal delays, Europhys. Lett. 79, 38001/1-5 (2007). Chapter 4: U. Meyer, J. Shao, S. Chakrabarty, S. F. Brandt, H. Luksch, and R. Wessel, Distributed delays stabilize neural feedback systems (submitted). arxiv: [physics.bio-ph]. Chapter 5: M. Caudill, S. F. Brandt, and R. Wessel, Dynamics of neural feedback triads with delays (in preparation). Chapter 6: S. F. Brandt and R. Wessel, Winner-take-all selection in a neural system with delayed feedback, Biol. Cybern. 97, (2007). Chapter 7: S. F. Brandt and R. Wessel, The isthmotectal feedback loop as a winner-take-all and novelty detection circuit (in preparation). Chapter 8: S. F. Brandt, A. Pelster, and R. Wessel, Noise-dependent stability of the synchronized state in a coupled system of active rotators (in preparation). Chapter 9: S. F. Brandt, B. K. Dellen, and R.Wessel, Synchronization from disordered driving forces in arrays of coupled oscillators, Phys. Rev. Lett. 96, /1-4 (2006).

3 Chapters 2 and 3 S. F. Brandt, A. Pelster, and R. Wessel, Variational calculation of the limit cycle and its frequency in a two-neuron model with delay, Phys. Rev. E 74, /1-14 (2006). S. F. Brandt, A. Pelster, and R. Wessel, Synchronization in a neuronal feedback loop through asymmetric temporal delays, Europhys. Lett. 79, 38001/1-5 (2007). τ 1 u ( ) u ( ) 2 t 1 t τ 2

4 Chapter 4 U. Meyer, J. Shao, S. Chakrabarty, S. F. Brandt, H. Luksch, and R. Wessel, Distributed delays stabilize neural feedback systems (submitted). arxiv: [physics.bio-ph]. τ 1 u ( ) u ( ) 2 t 1 t τ 2

5 Chapter 5 M. Caudill, S. F. Brandt, and R. Wessel, Dynamics of neural feedback triads with delays (in preparation).

6 Chapters 6 and 7 S. F. Brandt and R. Wessel, Winner-take-all selection in a neural system with delayed feedback, Biol. Cybern. 97, (2007). S. F. Brandt and R. Wessel, The isthmotectal feedback loop as a winner-take-all and novelty detection circuit (in preparation). Wang et al. (2004)

7 Chapter 8 S. F. Brandt, A. Pelster, and R. Wessel, Noise-dependent stability of the synchronized state in a coupled system of active rotators (in preparation).

8 Chapter 9 S. F. Brandt, B. K. Dellen, and R.Wessel, Synchronization from disordered driving forces in arrays of coupled oscillators, Phys. Rev. Lett. 96, /1-4 (2006).

9 Outline of Talk Chapter 2: S. F. Brandt, A. Pelster, and R. Wessel, Variational calculation of the limit cycle and its frequency in a two-neuron model with delay, Phys. Rev. E 74, /1-14 (2006). Chapter 3: S. F. Brandt, A. Pelster, and R. Wessel, Synchronization in a neuronal feedback loop through asymmetric temporal delays, Europhys. Lett. 79, 38001/1-5 (2007). u ( ) u ( t 2 ) 1 t τ 1 τ 2 Chapter 6: S. F. Brandt and R. Wessel, Winner-take-all selection in a neural system with delayed feedback, Biol. Cybern. 97, (2007). Chapter 7: S. F. Brandt and R. Wessel, The isthmotectal feedback loop as a winner-take-all and novelty detection circuit (in preparation).

10 Time-Continuous Model of Neuron Dynamics by Hopfield Leaky neuron with external input and input from other neurons: Hopfield (1984) : input voltage, : output voltage, : synaptic interconnection matrix Nonlinear Transfer Function:

11 Two-Neuron Model With Delay Model Equations: Characteristic Equation: Supercritical Hopf-Bifurcation:

12 Poincaré-Lindstedt Method Expansions: plane: Rescaling:

13 Perturbative Results Angular Frequency Limit Cycle

14 Basic Principles of Variational Perturbation Theory Divergent Weak Coupling Series Convergent Strong Coupling Expansion Example: quantum-mechanical anharmonic oscillator:

15 Example: Anharmonic Oscillator Weak Coupling Series for Ground-State Energy Identity: Substitution: Example: First Order

16 Principle of Minimal Sensitivity Conditions:

17 Strong-Coupling Limit : Results Exponential Convergence:

18 Perturbation Expansion: Angular Frequency Introduction of Variational Parameter to the Perturbation Expansion: First Order: Result:

19 VPT Results: Convergence:

20 Limit Cycle Fourier Series: Introduction of Variational Parameter to the Fourier Series: Evaluation for Optimal Value from Angular Frequency:

21 VPT Results: Convergence:

22 Comparison VPT/Shohat Brandt, Pelster, and Wessel (2006) Conduction Velocity Asymmetry in the Biological System:

23 Numerical Results Delay Parameter: Covariance:

24 First Order VPT

25 VPT Results Brandt, Pelster, and Wessel (2007)

26 Outline of Talk Chapter 2: S. F. Brandt, A. Pelster, and R. Wessel, Variational calculation of the limit cycle and its frequency in a two-neuron model with delay, Phys. Rev. E 74, /1-14 (2006). Chapter 3: S. F. Brandt, A. Pelster, and R. Wessel, Synchronization in a neuronal feedback loop through asymmetric temporal delays, Europhys. Lett. 79, 38001/1-5 (2007). u ( ) u ( t 2 ) 1 t τ 1 τ 2 Chapter 6: S. F. Brandt and R. Wessel, Winner-take-all selection in a neural system with delayed feedback, Biol. Cybern. 97, (2007). Chapter 7: S. F. Brandt and R. Wessel, The isthmotectal feedback loop as a winner-take-all and novelty detection circuit (in preparation).

27 Bottom-Up Model For Attention (Koch and Ullman, 1985) Itti and Koch (2001)

28 The Isthmotectal feedback loop as a WTA circuit Wang et al. (2004)

29 Model Rate Model (Hopfield, 1984): Assumptions: Membrane Time Constants: Delays: Transfer Functions:

30 Topology: System of 2N + 1First-Order DDE s: Weights:

31 Numerical Results for Local Excitation and Global Inhibition No delay With delay

32 Numerical Results for Local Inhibition and Global Excitation No delay With delay

33 Outline of Talk Chapter 2: S. F. Brandt, A. Pelster, and R. Wessel, Variational calculation of the limit cycle and its frequency in a two-neuron model with delay, Phys. Rev. E 74, /1-14 (2006). Chapter 3: S. F. Brandt, A. Pelster, and R. Wessel, Synchronization in a neuronal feedback loop through asymmetric temporal delays, Europhys. Lett. 79, 38001/1-5 (2007). u ( ) u ( t 2 ) 1 t τ 1 τ 2 Chapter 6: S. F. Brandt and R. Wessel, Winner-take-all selection in a neural system with delayed feedback, Biol. Cybern. 97, (2007). Chapter 7: S. F. Brandt and R. Wessel, The isthmotectal feedback loop as a winner-take-all and novelty detection circuit (in preparation).

34 Results from Marín et. al: Novelty Detection Marín et al., J. Neurosci. (2007)

35 Model Network of Integrate-and-Fire Neurons dv τ dt = V E ) ( L + R I V = V T : V V R

36 Model Network of Integrate-and-Fire Neurons dv τ dt = V E ) ( L + R I V = V T : V V R

37 Model Network of Integrate-and-Fire Neurons dv τ dt = V E ) ( L + R I V = V T : V V R

38 Model Parameters variable constant

39 Random Scan of Parameter Space Simulate 1000 networks with random combinations of parameters Score networks according to winner-take-all selection and novelty detection TeO Ipc Imc

40 - Best network: unchanged Genetic Algorithm - Top 50 % of networks percentile rank 10%: probability of1/30 for change of every parameter: 1/2top, 1/2 arithmetic mean 10% percentile rank < 20% probability of 1/15 for change of every parameter: 1/2top, 1/2 arithmetic mean 20% percentile rank < 30%: probability of 2/15 for change of every parameter: 1/2top, 1/2 arithmetic mean 30 % percentile rank < 50 % probability of1/3 for change of every parameter: 1/2top, 1/2 arithmetic mean Point mutation (+/- 1) with probability 1/10 - Bottom 50 % of networks With probability 1/3: all parameters: 1/2top, 1/2 arithmetic mean With probability 1/3: all parameters: 1/2 2nd best, 1/2 arithmetic mean With probability 1/3: all parameters: 1/2 3rd best, 1/2 arithmetic mean Point mutation (any value) with probability 1/10

41 Parameter Optimization Simulate 20 generations of 50 networks Use genetic algorithm to optimize for winner-take-all selection and novelty detection TeO Ipc Imc

42 Convergence of Genetic Algorithm Score of best three networks vs. generation #

43 Predicting Parameter Relations Measured Delays: Meyer et al. (submitted)

44 Summary Variational Resummation Synchrony from asymmetric delays Delays and WTA Parameter optimization through genetic algorithm