On the Dynamics of Delayed Neural Feedback Loops Sebastian Brandt Department of Physics, Washington University in St. Louis
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, 036201/1-14 (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:0712.0036 [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, 221-228 (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, 034104/1-4 (2006).
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, 036201/1-14 (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
Chapter 4 U. Meyer, J. Shao, S. Chakrabarty, S. F. Brandt, H. Luksch, and R. Wessel, Distributed delays stabilize neural feedback systems (submitted). arxiv:0712.0036 [physics.bio-ph]. τ 1 u ( ) u ( ) 2 t 1 t τ 2
Chapter 5 M. Caudill, S. F. Brandt, and R. Wessel, Dynamics of neural feedback triads with delays (in preparation).
Chapters 6 and 7 S. F. Brandt and R. Wessel, Winner-take-all selection in a neural system with delayed feedback, Biol. Cybern. 97, 221-228 (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)
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, 034104/1-4 (2006).
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, 036201/1-14 (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, 221-228 (2007). Chapter 7: S. F. Brandt and R. Wessel, The isthmotectal feedback loop as a winner-take-all and novelty detection circuit (in preparation).
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:
Two-Neuron Model With Delay Model Equations: Characteristic Equation: Supercritical Hopf-Bifurcation:
Poincaré-Lindstedt Method Expansions: plane: Rescaling:
Perturbative Results Angular Frequency Limit Cycle
Basic Principles of Variational Perturbation Theory Divergent Weak Coupling Series Convergent Strong Coupling Expansion Example: quantum-mechanical anharmonic oscillator:
Example: Anharmonic Oscillator Weak Coupling Series for Ground-State Energy Identity: Substitution: Example: First Order
Principle of Minimal Sensitivity Conditions:
Strong-Coupling Limit : Results Exponential Convergence:
Perturbation Expansion: Angular Frequency Introduction of Variational Parameter to the Perturbation Expansion: First Order: Result:
VPT Results: Convergence:
Limit Cycle Fourier Series: Introduction of Variational Parameter to the Fourier Series: Evaluation for Optimal Value from Angular Frequency:
VPT Results: Convergence:
Comparison VPT/Shohat Brandt, Pelster, and Wessel (2006) Conduction Velocity Asymmetry in the Biological System:
Numerical Results Delay Parameter: Covariance:
First Order VPT
VPT Results Brandt, Pelster, and Wessel (2007)
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, 036201/1-14 (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, 221-228 (2007). Chapter 7: S. F. Brandt and R. Wessel, The isthmotectal feedback loop as a winner-take-all and novelty detection circuit (in preparation).
Bottom-Up Model For Attention (Koch and Ullman, 1985) Itti and Koch (2001)
The Isthmotectal feedback loop as a WTA circuit Wang et al. (2004)
Model Rate Model (Hopfield, 1984): Assumptions: Membrane Time Constants: Delays: Transfer Functions:
Topology: System of 2N + 1First-Order DDE s: Weights:
Numerical Results for Local Excitation and Global Inhibition No delay With delay - - + + + - - - -
Numerical Results for Local Inhibition and Global Excitation No delay With delay + + - - - + + + +
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, 036201/1-14 (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, 221-228 (2007). Chapter 7: S. F. Brandt and R. Wessel, The isthmotectal feedback loop as a winner-take-all and novelty detection circuit (in preparation).
Results from Marín et. al: Novelty Detection 2 1 1 1 1 2 2 2 Marín et al., J. Neurosci. (2007)
Model Network of Integrate-and-Fire Neurons dv τ dt = V E ) ( L + R I V = V T : V V R
Model Network of Integrate-and-Fire Neurons dv τ dt = V E ) ( L + R I V = V T : V V R
Model Network of Integrate-and-Fire Neurons dv τ dt = V E ) ( L + R I V = V T : V V R
Model Parameters variable constant
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
- 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
Parameter Optimization Simulate 20 generations of 50 networks Use genetic algorithm to optimize for winner-take-all selection and novelty detection TeO Ipc Imc
Convergence of Genetic Algorithm Score of best three networks vs. generation #
Predicting Parameter Relations Measured Delays: Meyer et al. (submitted)
Summary Variational Resummation Synchrony from asymmetric delays Delays and WTA Parameter optimization through genetic algorithm