Part I: Single Neuron Models BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002 Chapters 2-5 Laboratory of Computational Neuroscience, LCN, CH 1015 Lausanne Swiss Federal Institute of Technology Lausanne, EPFL

Chapter 2: Detailed neuron models Hodgkin-Huxley model BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002 Chapter 2

Molecular basis action potential Ca 2+ Na + K+K+ -70mV Ions/proteins

Hodgkin-Huxley Model 100 mV 0 stimulus inside outside Ka Na Ion channels Ion pump Cglgl gKgK g Na I

Hodgkin-Huxley Model stimulus inside outside Ka Na Ion channels Ion pump u u h 0 (u) m 0 (u) pulse input I(t)

Hodgkin-Huxley Model 100 mV 0 020ms Strong stimulus strong stimuli Action potential refractoriness

Hodgkin-Huxley Model I(t) 100 mV 0 Stimulation with time-dependent input current

Hodgkin-Huxley Model I(t) mV mV 0 Subthreshold response Spike

Laboratory of Computational Neuroscience, LCN, CH 1015 Lausanne computational model Swiss Federal Institute of Technology Lausanne, EPFL neurons molecules ion channels signals From molecules to neuron models Ion channel based model justify

    Kv  1 Kv  2Kv  3 Kv  Voltage Activated K + Channels + + Kv1.1 Kv1.2Kv1.3Kv1.4Kv1.5 Kv1.6 Kv2.2Kv2.1 Kv3.2 Kv3.1 Kv3.3 Kv3.4 KChIP1 KChIP2KChIP3 KChIP Kv1Kv3Kv2Kv4Kv5Kv6Kv8Kv9 Kv Kv4.3 Kv4.1 Kv4.2 Ion Channels investigated in the study of CGRP CB PV CR NPY VIP SOM CCK pENK Dyn SP CRH schematic mRNA Expression profile Cglgl g Kv1 g Na I g Kv3

Cglgl g Kv1 g Na I g Kv3 Model of fast spiking interneuron stimulus inside outside Ka Na Ion channels Ion pump Erisir et al, 1999 Hodgkin and Huxley, 1952

Cglgl g Kv1 g Na I g Kv3 Model of fast spiking interneuron I(t) Spike Subthreshold Detailed model, based on ion channels

Cglgl g Kv1 g Na I g Kv3 Model of fast spiking interneuron Detailed model, based on ion channels Current pulse constant current

Chapter 3: Two-dimensional neuron models BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002 Chapter 3

Reduction of Hodgkin-Huxley Model stimulus u u h 0 (u) m 0 (u) 1) dynamics of m is fast 2) dynamics of h and n is similar

Reduction of Hodgkin-Huxley Model: 2 dimensional Neuron Models stimulus w u I(t)=I 0

FitzHugh-Nagumo Model 2-dimensional Neuron Models stimulus w u I(t)=0

FitzHugh Nagumo Model stimulus w u I(t)=I 0 limit cycle -unstable fixed point -closed boundary with arrows pointing inside limit cycle

type II Model stimulus w u I(t)=I 0 I0I0 Discontinuous gain function

type I Model stimulus w u I(t)=I 0 I0I0 Low-frequency firing

Chapter 4: Formal Spiking Neuron models Integrate-and-fire and SRM BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002 Chapter 4

Integrate-and-fire Model i Spike reception: EPSP Fire+reset linear threshold Spike emission reset I j

Nonlinear Integrate-and-fire Model i Fire+reset NONlinear threshold Spike emission reset I j F

Nonlinear Integrate-and-fire Model Fire+reset NONlinear threshold I<I u I>I u Quadratic I&F: arbitrary F(u)

I&F with time-dependent time constant i Fire+reset linear, time-dependent threshold Spike emission Soft reset I j

Spike Response Model i j Spike reception: EPSP Spike emission: AP Firing: linear threshold Spike emission Last spike of i All spikes, all neurons

Full Spike Response Model Firing: threshold Spike emission Last spike of i Volterra expansion potential Last spike of i

Integrate-and-fire Models Last spike of i LIF NLIF LIF:time dep. SRM 0 Last spike of i SRM

Integrate-and-fire Models Last spike of i LIF SRM 0 LIF is special case of SRM 0 Last spike of i

Integrate-and-fire Models Last spike of i LIF SRM Time-dep.LIF is special case of SRM Last spike of i

Comparison: Hodgkin-Huxley and SRM

Hodgkin-Huxley Model 100 mV 0 Example: refractoriness stimulus inside outside Ka Na Ion channels Ion pump

Full Spike Response Model Last spike of i Firing: threshold potential Hodgkin-Huxley Model vs. SRM 100 mV 0 020ms Spike emission

Full Spike Response Model Firing: threshold Last spike of i potential Hodgkin-Huxley Model 100 mV 0 020ms Hodgkin-Huxley Model vs. SRM 0 1 mV

Full Spike Response Model Hodgkin-Huxley Model I(t) mV mV 0 90% of firing times correctly predicted Kistler et al., 1997

Comparison: Detailed model of Fast-spiking interneuron vs. nonlinear I&F or SRM

Cglgl g Kv1 g Na I g Kv3 Validation of neuron models I(t) Spike threshold model

Cglgl g Kv1 g Na I g Kv3 Model of fast spiking interneuron stimulus inside outside Ka Na Ion channels Ion pump Erisir et al, 1999 Hodgkin and Huxley, 1952

Cglgl g Kv1 g Na I g Kv3 Model of fast spiking interneuron I(t) Spike Subthreshold Detailed model, based on ion channels

Cglgl g Kv1 g Na I g Kv3 Validation of neuron models I(t) Spike threshold model

Reduction of interneuron model To Nonlinear I&F stimulus Step 1 keep all ion currents – add threshold Fire+reset Reset – but to which value?

Reduction Step 1 keep all ion currents – add threshold constant current Pulse input I(t)

Reduction of interneuron model Step 2 separate in fast and slow ion currents Fire+reset Fast: Slow

Reduction of interneuron model Step 2 separate in fast and slow ion currents Fire+reset

Reduction Step 2 separate in fast and slow ion currents Pulse input I(t) Constant current

Cglgl g Kv1 g Na I g Kv3 Validation of neuron models I(t) u

Comparison interneuron model vs NONlinear Integrate-and-fire (numerical) Fire+reset I(t)

Cglgl g Kv1 g Na I g Kv3 Reduction of detailed model to SRM Spike I(t) u x subthreshold threshold

Reduction of interneuron model To Spike Response Model Fire+reset NONlinear threshold Spike emission

Reduction of interneuron model To Spike Response Model Step 1 analyze behavior after a spike

Reduction of interneuron model To Spike Response Model Fire+reset time dependent threshold Spike emission

Reduction To SRM constant current Pulse input I(t)

C glgl g Kv1 g Na I g Kv3 Comparison: detailed vs SRM I(t) detailed model Spike threshold model (SRM) <2ms 80% of spikes correct (+/-2ms)

Comparison: interneuron model vs Spike Response Model (numerical) Fire+reset I(t)

Cglgl g Kv1 g Na I g Kv3 Validation of neuron models I(t) Spike threshold model 80 % of spikes correct 75 % of spikes correct

Chapter 5: Noise in Spiking Neuron Models Escape noise (noisy threshold) Slow noise in parameters (adiabatic noise) Diffusive noise (stochastic spike arrival) BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002 Chapter 5

Variability of spike trains =noise? ISI distribution t [ms] ISI u [mV] time [ms] 50 - Variability of interspike intervals t - Variability across repetitions K repetitions

Variability of spike trains =noise? sources of noise - Intrinsic noise (ion channels) -Network noise (background activity) Ca 2+ Na + K+K+

escape process (fast noise) parameter changes (slow noise) stochastic spike arrival (diffusive noise) Noise models AB C u(t) noise white (fast noise) synapse (slow noise) (Brunel et al., 2001) : first passage time problem Interval distribution t Survivor function escape rate escape rate stochastic reset Interval distribution Gaussian about noisy integration

Escape Noise (noisy threshold) escape rate Example 1): neuron with absolute refractoriness, constant input Survivor function 1 Interval distribution

Escape Noise (noisy threshold) escape rate Example 2): I&F with reset, constant input, exponential escape rate Survivor function 1 Interval distribution

Stochastic spike arrival: excitation, total rate R e inhibition, total rate R i u EPSC IPSC current pulses Diffusive noise (stochastic spike arrival)

Stochastic spike arrival: excitation, total rate R e inhibition, total rate R i u EPSC IPSC Synaptic current pulses Diffusive noise (stochastic spike arrival) Langenvin equation, Ornstein Uhlenbeck process

u p(u) Diffusive noise (stochastic spike arrival) u Membrane potential density Fokker-Planck drift diffusion spike arrival rate

u p(u) Diffusive noise (stochastic spike arrival) u Membrane potential density: Gaussian Fokker-Planck drift diffusion spike arrival rate constant input rates no threshold

u p(u) Diffusive noise + Threshold u Membrane potential density p(u) Hypothetical experiment: voltage step Immediate response vanishes quadratically Fokker-Planck Density at threshold p=0

Diffusive noise (stochastic spike arrival) Superthreshold vs. Subthreshold regime

stochastic spike arrival (diffusive noise) Noise models: from diffusive noise to escape rates escape rate noisy integration

Comparison: diffusive noise vs. escape rates escape rate subthreshold potential Probability of first spike diffusive escape