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Dynamical Systems Analysis for Systems of Spiking Neurons

Dynamical Systems Analysis for Systems of Spiking Neurons. Models: Leaky Integrate and Fire Model. CdV/dt= -V/R+I syn Resting Potential V Rest assumed to be 0. CR = Membrane time constant (20 msec for excitatory neurons, 10 msec for inhibitory neurons.)

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Dynamical Systems Analysis for Systems of Spiking Neurons

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  1. Dynamical Systems Analysis for Systems of Spiking Neurons

  2. Models:Leaky Integrate and Fire Model • CdV/dt= -V/R+Isyn • Resting Potential VRest assumed to be 0. • CR = Membrane time constant (20 msec for excitatory neurons, 10 msec for inhibitory neurons.) • Spike generated when V reaches VThreshold • Voltage reset to VResetafter spike (not the same as VRest) • Synaptic Current Isyn assumed to be either delta function or alpha function.

  3. Models:Spike-Response Model Observation: The L-IF-model is linear CdV1/dt= -V1/R+I1syn CdV2/dt= -V2/R+I2syn Cd(V1+V2)/dt= -(V1+V2)/R+I1syn+I2syn Why not simply take the individual effect of each spike and add them all up? Result: The Spike response model. V(t)=effect of previously generated spikes by neuron+ sum over all effects generated by spikes that have arrived at synapses

  4. Synapse • Dendrites (Input) • Cell Body • Axon (Output) Background:The Cortical Neuron Output Input Threshold • Absolute Refractory Period • Exponential Decay of effect of a spike on membrane potential Time

  5. Background:Target System Neocortical Column: ~ 1 mm2 of the cortex Recurrent network ~100,000 neurons ~10,000 synapses per neuron ~80% excitatory ~20% inhibitory Output Recurrent System Input

  6. Background:The Neocortex (Healthy adult human male subject) Source: Dr. Krishna Nayak, SCRI, FSU

  7. Background: The Neocortex (Area V1 of Macaque Monkey) Source: Dr. Wyeth Bair, CNS, NYU

  8. Background:Dynamical Systems Analysis • Phase Space • Set of all legal states • Dynamics • Velocity Field • Flows • Mapping • Local & Global properties • Sensitivity to initial conditions • Fixed points and periodic orbits

  9. Content: • Model • A neuron • System of Neurons: Phase Space&Velocity Field • Simulation Experiments • Neocortical Column • Qualitative Characteristics: EEG power spectrum&ISI frequency distribution • Formal Analysis • Local Analysis: Sensitivity to Initial Conditions • Conclusions

  10. t=0 t=0 t=0 Time Model:Single Neuron Potential Function Each spike represented as:How long since it departed from soma.

  11. Model:Single Neuron: Potential function

  12. Model:System of Neurons • Dynamics • Birth of a spike • Death of a spike • Point in the Phase-Space • Configuration of spikes

  13. Model:Single Neuron: Phase-Space

  14. Model:Single Neuron: Phase-Space Theorem: Phase-Space can be defined formally Phase-Space for Total Number of Spikes Assigned = 1, 2, & 3.

  15. Model:Single Neuron: Structure of Phase-Space • Phase-Space for n=3 • 1, 2 dead spikes.

  16. Model:System of Neurons: Velocity Field

  17. Simulations: Neocortical Column: Setup • 1000 neurons each connected randomly to 100 neurons. • 80% randomly chosen to be excitatory, rest inhibitory. • Basic Spike-response model. • Total number of active spikes in the system ►EEG / LFP recordings • Spike Activity of randomly chosen neurons ►Real spike train recordings • 5 models: Successively enhanced physiological accuracy • Simplest model • Identical EPSPs and IPSPs, IPSP 6 times stronger • Most complex model • Synapses: Excitatory (50% AMPA, NMDA), Inhibitory (50% GABAA, GABAB) • Realistic distribution of synapses on soma and dendrites • Synaptic response as reported in (Bernander Douglas & Koch 1992)

  18. Simulations: Neocortical Column: Classes of Activity Number of active spikes: Seizure-like & Normal Operational Conditions

  19. Simulations: Neocortical Column: Chaotic Activity T=0 T=1000 msec Normal Operational Conditions (20 Hz):Subset (200 neurons) of 1000 neurons for 1 second.

  20. Simulations: Neocortical Column: Total Activity Normalized time series:Total number of active spikes & Power Spectrum

  21. Simulations: Neocortical Column: Spike Trains Representative spike trains: Inter-spike Intervals & Frequency Distributions

  22. Simulations: Neocortical Column: Propensity for Chaos ISI’s of representative neurons: 3 systems; 70%,80%,90% synapses driven by pacemaker

  23. Simulations: Neocortical Column: Sensitive Dependence on Initial Conditions T=0 T=400 msec Spike activity of 2 Systems: Identical Systems, subset (200) of 1000 neurons, Identical Initial State except for1 spike perturbed by 1 msec.

  24. Analysis: Local Analysis • Are trajectories sensitive to initial conditions? • If there are fixed points or periodic orbits, are they stable?

  25. Analysis: Setup: Riemannian Metric

  26. Analysis: Setup: Riemannian Metric • Discrete Dynamical System • Event ► Event ►Event…. • Event: birth/death of spike

  27. Analysis: Measure Analysis Death of a Spike PI Birth of a Spike

  28. Analysis: Perturbation Analysis

  29. Analysis: Perturbation Analysis

  30. Birth Death If then sensitive to initial conditions. If then insensitive to initial conditions. Analysis: Local Cross-Section Analysis AT B C

  31. Analysis: Local Cross-Section Analysis

  32. Analysis: Local Cross-Section Analysis: Prediction

  33. Seizure Normal Spike rate =1 >1 =1 <1 Analysis: Local Cross-Section Analysis: Prediction Neocortical Column

  34. Analysis: Discussion • Existence of time average • Systems without Input and with Stationary Input • Transformation invariant (Stationary) Probability measure exists. • System has Ergodic properties. • Systems with Transient Inputs • ? • InformationCoding (Computational State vs. Physical State) • Attractor-equivalent of class of trajectories.

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