Neuronal Dynamics: Computational Neuroscience of Single Neurons Week 7 – Optimizing Neuron Models For Coding and Decoding Wulfram Gerstner EPFL, Lausanne, Switzerland 7.1 What is a good neuron model? - Models and data 7.2 AdEx model - Firing patterns and analysis 7.3 Spike Response Model (SRM) - Integral formulation 7.4 Generalized Linear Model (GLM) - Adding noise to the SRM 7.5 Parameter Estimation - Quadratic and convex optimization 7.6. Modeling in vitro data - how long lasts the effect of a spike? 7.7. Helping Humans Week 7 – part 5 : Parameter estimation
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Neuronal Dynamics: Computational Neuroscience of Single Neurons Week 7 – Optimizing Neuron Models For Coding and Decoding Wulfram Gerstner EPFL, Lausanne, Switzerland
7.1 What is a good neuron model? - Models and data 7.2 AdEx model
- Firing patterns and analysis 7.3 Spike Response Model (SRM) - Integral formulation 7.4 Generalized Linear Model (GLM) - Adding noise to the SRM 7.5 Parameter Estimation - Quadratic and convex optimization 7.6. Modeling in vitro data - how long lasts the effect of a spike? 7.7. Helping Humans
Week 7 – part 5 : Parameter estimation
7.1 What is a good neuron model? - Models and data 7.2 AdEx model
- Firing patterns and analysis 7.3 Spike Response Model (SRM) - Integral formulation 7.4 Generalized Linear Model (GLM) - Adding noise to the SRM 7.5 Parameter Estimation - Quadratic and convex optimization 7.6. Modeling in vitro data - how long lasts the effect of a spike? 7.7. Helping Humans
Week 7 – part 5 : Parameter estimation
( )0
( ) rests I t s ds uκ∞
+ − +∫Subthreshold potential
( ) ( )s S t s dsη −∫( )=tu
Linear filters/linear in parameters known spike train known input
Neuronal Dynamics – 7.5 Parameter estimation: voltage
ih I(t)
( )sκ
( )sη
S(t)
1( )sθ
+ ( )f u ϑ−
Spike Response Model (SRM) Generalized Lin. Model (GLM)
( )n k n k restu t k I u−= +∑
Linear in parameters = linear fit = quadratic problem Neuronal Dynamics – 7.5 Parameter estimation: voltage
comparison model-data
( )0
( ) ( ) restu t s I t s ds uκ∞
= − +∫
( )0
( ) ( ) restu t s I t s ds uκ∞
= − +∫( )n k n k restu t k I u−= +∑
Linear in parameters = linear fit Neuronal Dynamics – 7.5 Parameter estimation: voltage
I
( )datau t
2
1[ ( ) ]
Kdata
n k n k restn k
E u t k I u−=
= − −∑ ∑
( )0
( ) ( ) restu t s I t s ds uκ∞
= − +∫( )n k n k rest
ku t k I u−= +∑
Model Data ( )datau t
I
( )datau t
2
1[ ( ) ]
Kdata
n k n k restn k
E u t k I u−=
= − −∑ ∑
Linear in parameters = linear fit = quadratic optimization Neuronal Dynamics – 7.5 Parameter estimation: voltage
( )n k n k restk
u t k I u−= +∑
2
1[ ( ) ]
Kdata
n k n k restn k
E u t k I u−=
= − −∑ ∑
( )n nu t k x= ⋅r r
Neuronal Dynamics – 7.5 Parameter estimation: voltage
Vector notation
( )n nu t k x= ⋅r r
( )0
( ) ( ) restu t s I t s ds uκ∞
= − +∫( )n k n k restu t k I u−= +∑
Linear in parameters = linear fit = quadratic problem
( )n nu t k x= ⋅r r
2
1[ ( ) ]
Kdata
n k n k restn k
E u t k I u−=
= − −∑ ∑
( )0
( )s S t s dsη∞
+ −∫
Neuronal Dynamics – 7.5 Parameter estimation: voltage
Mensi et al., 2012
( )0
( ) rests I t s ds uκ∞
− +∫Subthreshold potential
( ) ( )s S t s dsη+ −∫( )=tuknown spike train known input
pyramidal
inhibitory interneuron
pyramidal
Neuronal Dynamics – 7.5 Extracted parameters: voltage
A) Predict spike times B) Predict subthreshold voltage C) Easy to interpret (not a ‘black box’) D) Flexible E) Systematic: ‘optimize’ parameters
Neuronal Dynamics – What is a good neuron model?
7.1 What is a good neuron model? - Models and data 7.2 AdEx model
- Firing patterns and analysis 7.3 Spike Response Model (SRM) - Integral formulation 7.4 Generalized Linear Model (GLM) - Adding noise to the SRM 7.5 Parameter Estimation - Quadratic optimization: subthreshold - convex optimization: spike times 7.6. Modeling in vitro data - how long lasts the effect of a spike? 7.7. Helping Humans
Week 7 – part 5b : Parameter estimation for spike times
Neuronal Dynamics: Computational Neuroscience of Single Neurons Week 7 – Optimizing Neuron Models For Coding and Decoding Wulfram Gerstner EPFL, Lausanne, Switzerland
7.1 What is a good neuron model? - Models and data 7.2 AdEx model
- Firing patterns and analysis 7.3 Spike Response Model (SRM) - Integral formulation 7.4 Generalized Linear Model (GLM) - Adding noise to the SRM 7.5 Parameter Estimation - Quadratic optimization: subthreshold - convex optimization: spike times 7.6. Modeling in vitro data - how long lasts the effect of a spike? 7.7. Helping Humans
Week 7 – part 5b : Parameter estimation for spike times
Fitting models to data: so far ‘subthreshold’
Adaptation current
Dyn. threshold
( )0
( ) rests I t s ds uκ∞
+ − +∫potential ( ) ( )s S t s dsη −∫( )=tu
0 1( ) ( ) ( )t s S t s dsϑ θ θ= + −∫threshold
firing intensity ( ) ( ( ) ( ))t f u t tρ ϑ= −
ih I(t) ( )sκ
( )sη
S(t)
1( )sθ
+ ( )f u ϑ−
Jolivet&Gerstner, 2005 Paninski et al., 2004
Pillow et al. 2008
Neuronal Dynamics – 7.5 Threshold: Predicting spike times
( )0
( ) rests I t s ds uκ∞
+ − +∫potential ( ) ( )s S t s dsη −∫( )=tu
0 1( ) ( ) ( )t s S t s dsϑ θ θ= + −∫threshold
firing intensity ( ) ( ( ) ( ))t f u t tρ ϑ= −
1
0
log ( ,..., ) ( ') ' log ( )T
N f
fL t t t dt t Eρ ρ= − + = −∑∫
Neuronal Dynamics – 7.5 Generalized Linear Model (GLM)
( )0
( ) rests I t s ds uκ∞
+ − +∫potential ( ) ( )s S t s dsη −∫( )=tu
0 1( ) ( ) ( )t s S t s dsϑ θ θ= + −∫threshold
firing intensity ( ) ( ( ) ( ))t f u t tρ ϑ= −
1
0
log ( ,..., ) ( ') ' log ( )T
N f
fL t t t dt tρ ρ= − +∑∫
Paninski, 2004
Neuronal Dynamics – 7.5 GLM: concave error function
Neuronal Dynamics – 7.5 quadratic and convex/concave optimization
Voltage/subthreshold - linear in parameters à quadratic error function Spike times - nonlinear, but GLM à convex error function
