biological modeling of neural networks:

Biological Modeling of Neural Networks:

Week 9 – Optimizing Neuron Models

For Coding and Decoding

Wulfram GerstnerEPFL, Lausanne, Switzerland

9.1 What is a good neuron model? - Models and data9.2 AdEx model

- Firing patterns and adaptation9.3 Spike Response Model (SRM) - Integral formulation9.4 Generalized Linear Model - Adding noise to the SRM 9.5 Parameter Estimation - Quadratic and convex optimization9.6. Modeling in vitro data - how long lasts the effect of a spike?

Week 9 – part 1 : Models and data

-What is a good neuron model?-Estimate parameters of models?

Neuronal Dynamics – 9.1 Neuron Models and Data

A) Predict spike timesB) Predict subthreshold voltageC) Easy to interpret (not a ‘black box’)D) Flexible enough to account for a variety of phenomenaE) Systematic procedure to ‘optimize’ parameters

Neuronal Dynamics – 9.1 What is a good neuron model?

( ) ( )du

f u R I tdt

What is a good choice of f ?

reset

r

If u

then reset to

u u

ru

Neuronal Dynamics – Review: Nonlinear Integrate-and-fire

See:week 1,lecture 1.5

( ) ( )du

f u R I tdt

What is a good choice of f ?

(i) Extract f from more complex models

(ii) Extract f from data

reset rIf u then reset to u u (2)

(1)


( ) ( )du

f u R I tdt


( , ) ( )du

F u w R I tdt

),( wuGdt

dww

See week 3:2dim version of Hodgkin-Huxley

Separation of time scales:Arrows are nearly horizontal

resting state

restw wSpike initiation, from rest

A. detect spike and reset

B. Assume w=wrest



( , ) ( )du

F u w R I tdt

),( wuGdt

dww

Separation of time scalesrestw w

( , ) ( )rest

duF u w R I t

dt

linear exponential

See week 4:2dim version of Hodgkin-Huxley


( ) ( )du

f u R I tdt

(ii) Extract f from data

Pyramidal neuron Inhibitory interneuron

( )( )

f uf u

linear exponential

Badel et al. (2008)

Badel et al. (2008)

( ) exp( )rest

du uu u

dt

Exp. Integrate-and-Fire, Fourcaud et al. 2003


( ) ( )du

f u R I tdt

linear exponential

( ) ( )du

f u R I tdt

Best choice of f : linear + exponential

reset rIf u then reset to u u

(1)

(2)

( ) exp( )rest

du uu u

dt

BUT: Limitations – need to add

-Adaptation on slower time scales-Possibility for a diversity of firing patterns-Increased threshold after each spike-Noise








Week 9 – part 2 : Adaptive Expontential Integrate-and-Fire Model

I(t)

Step current input – neurons show adaptation

1-dimensional (nonlinear) integrate-and-fire model cannot do this!

Data: Markram et al. (2004)

Neuronal Dynamics – 9.2 Adaptation

( ) exp( ) ( )rest kk

du uu u R w RI t

dt

Add adaptation variables:

( ) ( )fkk k rest k k k f

dwa u u w b t t

dt

kw

jumps by an amount kbafter each spike


SPIKE ANDRESET

AdEx model,Brette&Gerstner (2005):

Neuronal Dynamics – 9.2 Adaptive Exponential I&F

Exponential I&F+ 1 adaptation var.= AdEx

Blackboard !

Firing patterns:Response to Step currents,Exper. Data,Markram et al. (2004)I(t)

Firing patterns:Response to Step currents,AdEx Model,Naud&Gerstner

I(t)

Image:Neuronal Dynamics,Gerstner et al.Cambridge (2002)

( ) exp( ) ( )rest

du uu u Rw RI t

dt

( ) ( )fw rest w f

dwa u u w b t t

dt

AdEx model

Can we understand the different firing patterns?

Phase plane analysis!

Neuronal Dynamics – 9.2 Adaptive Exponential I&F

( ) exp( ) ( )rest

du uu u Rw RI t

dt

( )w rest

dwa u u w

dt

A - What is the qualitative shape of the w-nullcline? [ ] constant [ ] linear, slope a [ ] linear, slope 1 [ ] linear + quadratic [ ] linear + exponential

Neuronal Dynamics – Quiz 9.1. Nullclines of AdEx

B - What is the qualitative shape of the u-nullcline? [ ] linear, slope 1 [ ] linear, slope 1/R [ ] linear + quadratic [ ] linear w. slope 1/R+ exponential

1 minuteRestart at 9:38







Week 9 – part 2b : Firing Patterns

( ) exp( ) ( )rest

du uu u Rw RI t

dt

( ) ( )fw rest w f

dwa u u w b t t

dt

AdEx model

w jumps by an amount b after each spike

u is reset to ur

after each spike

parameter a – slope of w-nullcline

Can we understand the different firing patterns?

( ) exp( ) ( )rest

du uu u w RI t

dt

( ) ( )fw rest w f

dwa u u w b t t

dt

AdEx model – phase plane analysis: large b

b

u-nullcline

u is reset to ur

a=0

AdEx model – phase plane analysis: small b

b

u-nullcline

u is reset to ur

adaptation

( ) exp( ) ( )rest

du uu u w RI t

dt

( ) ( )fw rest w f

dwa u u w b t t

dt

( ) exp( ) ( )rest

du uu u w RI t

dt

( ) ( )fw rest w f

dwa u u b t t

dt

Quiz 9.2: AdEx model – phase plane analysis

b

u-nullcline

u is reset to ur

What firing pattern do you expect?(i) Adapting(ii) Bursting(iii) Initial burst(iv)Non-adapting

b

u-nullcline

u is reset to ur

( ) exp( ) ( )rest

du uu u w RI t

dt

( ) ( )fw rest w f

dwa u u w b t t

dt

AdEx model – phase plane analysis: a>0

( ) exp( ) ( )rest

du uu u Rw RI t

dt

( ) ( )fw rest w f

dwa u u w b t t

dt

Firing patterns arise from different parameters!

w jumps by an amount b after each spike

u is reset to urafter each spike

parameter a – slope of w nullcline

See Naud et al. (2008), see also Izikhevich (2003)

Neuronal Dynamics – 9.2 AdEx model and firing patterns

( ) ( )du

f u R I tdt

Best choice of f : linear + exponential


(1)

(2)

( ) exp( )rest

du uu u

dt

BUT: Limitations – need to add

-Adaptation on slower time scales-Possibility for a diversity of firing patterns-Increased threshold after each spike-Noise


( ) exp( ) ( )rest kk

du uu u R w RI t

dt

Add dynamic threshold:

0 1( )fft t

Threshold increases after each spike

Neuronal Dynamics – 9.2 AdEx with dynamic threshold

( ) ( )du

f u R I tdt


add-Adaptation variables-Possibility for firing patterns-Dynamic threshold -Noise

Neuronal Dynamics – 9.2 Generalized Integrate-and-fire







Week 9 – part 3: Spike Response Model (SRM)

( ) exp( ) ( )rest

du uu u RI t

dt

Exponential versus Leaky Integrate-and-Fire

( ) ( )rest

duu u RI t

dt

Reset if u=

Badel et al (2008)

2mV

Leaky Integrate-and-Fire

( ) ( )rest kk

duu u R w RI t

dt


dwa u u w b t t

dt

kw jumps by an amount kbafter each spike

( ) rIf u t then reset to u u

SPIKE ANDRESET

Dynamic threshold

Neuronal Dynamics – 9.3 Adaptive leaky integrate-and-fire

( ) ( )rest kk

duu u R w RI t

dt


dwa u u w b t t

dt

Linear equation can be integrated!

0

( ) ( ) ( ) ( )f

fu t t t ds s I t s

0 1( ) ( )ff

t t t Spike Response Model (SRM)

Gerstner et al. (1996)

Neuronal Dynamics – 9.3 Adaptive leaky I&F and SRM

Adaptive leaky I&F

Spike emission

0

( ) rests I t s ds u

potential

''t

t t tu

0 1'

( ) ( ')t

t t t threshold

Arbitrary Linear filters

( )tiu

iInput I(t) ( )s

Gerstner et al.,1993, 1996

u(t)

Neuronal Dynamics – 9.3 Spike Response Model (SRM)

SRM with appropriate leads to bursting

0

( ) ( ) ( ) ( )frestf

u t t t ds s I t s u

Neuronal Dynamics – 9.3 Bursting in the SRM

0 0

( ) ( ) ( ) ( ) ( ) restu t ds s S t s ds s I t s u

( ) ( )rest

duu u w RI t

dt

( )fw w f

dww b t t

dt

Exercise 1: from adaptive IF to SRM

Integrate the above system of two differential equations so as to rewrite the equations as

0


potential 0

( )s S t s ds

tu

A – what is ?

B – what is ?

s

s

exp( )R s

x s

exp( )w w

R sx s

[exp( ) exp( )]w

s sx s C

(i) (ii)

(iii) (iv) Combi of (i) + (iii)

rIf u then reset to u u

Next lectureat 9:57/10:15

ih

0


potential ''t

t t tu

0 1'

( ) ( ')t

t t t threshold

Input I(t) ( )s

( )s

S(t)

1( )s

+u

firing if ( ) ( )u t t

Gerstner et al.,1993, 1996


+


Linear filters for - input - threshold - refractoriness

0


potential

''t

t t tu

0 1'

( ) ( ')t

t t t threshold







Week 9 – part 4: Generalized Linear Model (GLM)

Spike Response Model (SRM)Generalized Linear Model GLM

0


potential ( )s S t s ds tu

0 1( ) ( ) ( )t s S t s ds threshold

firing intensity ( ) ( ( ) ( ))t f u t t

Gerstner et al., 1992,2000Truccolo et al., 2005Pillow et al. 2008

ih I(t) ( )s

( )s

S(t)

1( )s

+( )f u

escape process

u(t)

t

)(t

))(()( tuftescape rate

u

Neuronal Dynamics – review from week 8: Escape noise

1 ( )( ) exp( )

u tt

escape rate

( ) ( )rest

du u u RI t

dt

f frif spike at t u t u

Example: leaky integrate-and-fire model

t̂

( ) ( )rest

duu u RI t

dt

Exerc. 2.1: Non-leaky IF with escape rates

Integrate for constant input (repetitive firing)

0rest rreset tou u

Next lectureat 10:38

1( ) ( )

du RI t I t

dt C

0rreset to u

Calculate - potential

- hazard

- survivor function

- interval distrib.

ˆ( )u t t

ˆ ˆ( ) [ ( ) ]t t u t t

ˆ( )S t t

0ˆ( )P t t

nonleaky

escape process

u(t)

t

t

dtt^

)')'(exp( )ˆ( ttS I

Survivor function

^t t

)(t

( ) ( ( ) )t f u t t escape rate

t

t

dttt^

)')'(exp()( )ˆ( ttPI

Interval distribution

Survivor functionescape rate

)ˆ()()ˆ( ttStttS IIdtd

u

0

( )( ) ( ( ) ) exp[ ]

u t tt f u t t

u

Good choice

Neuronal Dynamics – review from week 8: Escape noise

( ) ( )f

f

S t t t

1t 2t 3t

1 2, ,... Nt t tMeasured spike train with spike times

Likelihood L that this spike train could have been generated by model?

1 2

1

1 1

0

( ,..., ) exp( ( ') ') ( ) exp( ( ') ')...t t

N

t

L t t t dt t t dt

0 T

Neuronal Dynamics – 9.4 Likelihood of a spike train in GLMs

Blackboard

1 2

1

1 1 2

0

( ,..., ) exp( ( ') ') ( ) exp( ( ') ') ( )... exp( ( ') ')N

t t TN

t t

L t t t dt t t dt t t dt

( ) ( )f

f

S t t t

1t 2t 3t0 T

1

0

( ,..., ) exp( ( ') ') ( )T

N f

f

L t t t dt t

1

0

log ( ,..., ) ( ') ' log ( )T

N f

f

L t t t dt t

Neuronal Dynamics – 9.4 Likelihood of a spike train

+

Neuronal Dynamics – 9.4 SRM with escape noise = GLM

-linear filters-escape ratelikelihood of observed spike trainparameter optimization of neuron model







Week 9 – part 5: Parameter Estimation

0


Subthreshold potential

( )s S t s ds tu

Linear filters/linear in parameters

known spike train known input

Neuronal Dynamics – 9.5 Parameter estimation: voltage

ih

I(t)

( )s

( )s

S(t)

1( )s

+( )f u

Spike Response Model (SRM)Generalized Lin. Model (GLM)

( )n k n k restu t k I u

Linear in parameters = linear fit = quadratic problem


comparison model-data

0

( ) ( ) restu t s I t s ds u

Blackboard: Riemann-sum

0



Linear in parameters = linear fit


2

1

[ ( ) ]K

datan k n k rest

n k

E u t k I u

Blackboard: Error function

I

( )datau t

0


( )n k n k rest

k

u t k I u Model

Data ( )datau t

I

( )datau t

2

1

[ ( ) ]K

datan k n k rest

n k

E u t k I u

Linear in parameters = linear fit = quadratic optimization


Mensi et al., 2012

0


Subthreshold potential

( )s S t s ds tuknown spike trainknown input

pyramidal

inhibitoryinterneuron

pyramidal

Neuronal Dynamics – 9.5 Extracted parameters: voltage

n nu RI

Exercise 3 NOW: optimize 1 free parameter

Model Data( )datanu t

I

( )datau t

Optimize parameter R, so as to have a minimal error2[ ( ) ]data

n nn

E u t RI

A) Predict spike timesB) Predict subthreshold voltageC) Easy to interpret (not a ‘black box’)D) FlexibleE) Systematic: ‘optimize’ parameters

Neuronal Dynamics – What is a good neuron model?







Week 9 – part 5b: Quadratic and Convex Optimization

Fitting models to data: so far ‘subthreshold’

Adaptation current

Dyn. threshold

0





ih I(t) ( )s

( )s

S(t)

1( )s

+( )f u

Jolivet&Gerstner, 2005Paninski et al., 2004

Pillow et al. 2008

Neuronal Dynamics – 9.5 Threshold: Predicting spike times

0





1

0

log ( ,..., ) ( ') ' log ( )T

N f

f

L t t t dt t E Neuronal Dynamics – 9.5 Generalized Linear Model (GLM)

0





1

0

log ( ,..., ) ( ') ' log ( )T

N f

f

L t t t dt t

Paninski, 2004

Neuronal Dynamics – 9.5 GLM: concave error function

Neuronal Dynamics – 9.5 quadratic and convex/concave optimization

Voltage/subthreshold - linear in parameters quadratic error function

Spike times - nonlinear, but GLM convex error function

ih

0




I(t) ( )s

( )s

S(t)

1( )s

+( )f u


Quiz NOW :What are the units of ? (i) Voltage?

(ii) Resistance?(iii) Resistance/s?(iv)Current?

( )s

ih

0




I(t) ( )s

( )s

S(t)

1( )s

+( )f u


Quiz NOW:What are the units of ? (i) Voltage?

(ii) Resistance?(iii) Resistance/s?(iv)Current?

( )s






- Firing patterns and adaptation9.3 Spike Response Model (SRM) - Integral formulation9.4 Generalized Linear Model - Adding noise to the SRM 9.5 Parameter Estimation - Quadratic and convex optimization9.6. Modeling in vitro data - how long lasts the effect of a spike?9.7. Helping Humans

Week 9 – part 6: Modeling in vitro data

Neuronal Dynamics – 9.6 Models and Data

comparison model-data

Predict-Subthreshold voltage-Spike times

0





ih I(t) ( )s

( )s

S(t)

1( )s

+( )f u

Jolivet&Gerstner, 2005Paninski et al., 2004

Pillow et al. 2008

Neuronal Dynamics – 9.6 GLM/SRM with escape noise

neuron

model

Image:Mensi et al.

Neuronal Dynamics – 9.6 GLM/SRM predict subthreshold voltage

Mensi et al., 2012

No moving threshold

With moving threshold

Role of moving threshold

Neuronal Dynamics – 9.6 GLM/SRM predict spike times

( )I tpotential ( )s S t s ds dC u tdt



Change in model formulation:What are the units of …. ?

uI(t)

( )s

S(t)

1( )s

+( )f u ( )s

exponential

adaptationcurrent

‘soft-thresholdadaptive IF model’

Pozzorini et al. 2013

Time scale of filters? Power law

A single spike has a measurable effect more than 10 seconds later!

( )s 1( )s

Neuronal Dynamics – 9.6 How long does the effect of a spike last?

-Predict spike times-Predict subthreshold voltage-Easy to interpret (not a ‘black box’)-Variety of phenomena-Systematic: ‘optimize’ parameters

Neuronal Dynamics – 9.6 Models and Data

BUT so far limited to in vitro

Neuronal Dynamics week 7– Suggested Reading/selected referencesReading: W. Gerstner, W.M. Kistler, R. Naud and L. Paninski,

Neuronal Dynamics: from single neurons to networks and models of cognition. Ch. 6,10,11: Cambridge, 2014

-Brillinger, D. R. (1988). Maximum likelihood analysis of spike trains of interacting nerve cells. Biol. Cybern., 59:189-200.-Truccolo, et al. (2005). A point process framework for relating neural spiking activity to spiking history, neural ensemble, and extrinsic covariate effects. Journal of Neurophysiology, 93:1074-1089.- Paninski, L. (2004). Maximum likelihood estimation of … Network: Computation in Neural Systems, 15:243-262.- Paninski, L., Pillow, J., and Lewi, J. (2007). Statistical models for neural encoding, decoding, and optimalstimulus design. In Cisek, P., et al. , Comput. Neuroscience: Theoretical Insights into Brain Function. Elsevier Science.Pillow, J., ET AL.(2008). Spatio-temporal correlations and visual signalling… . Nature, 454:995-999.

Rieke, F., Warland, D., de Ruyter van Steveninck, R., and Bialek, W. (1997). Spikes - Exploring the neural code. MIT Press,Keat, J., Reinagel, P., Reid, R., and Meister, M. (2001). Predicting every spike … Neuron, 30:803-817.Mensi, S., et al. (2012). Parameter extraction and classication …. J. Neurophys.,107:1756-1775.Pozzorini, C., Naud, R., Mensi, S., and Gerstner, W. (2013). Temporal whitening by . Nat. Neuroscience,Georgopoulos, A. P., Schwartz, A.,Kettner, R. E. (1986). Neuronal population coding of movement direction. Science, 233:1416-1419.Donoghue, J. (2002). Connecting cortex to machines: recent advances in brain interfaces. Nat. Neurosci., 5:1085-1088.

Fourcaud-Trocme, N., Hansel, D., van Vreeswijk, C., and Brunel, N. (2003). How spike …. J. Neuroscience, 23:11628-11640.Badel, L., et al. (2008a). Extracting nonlinear integrate-and-fire, Biol. Cybernetics, 99:361-370.Brette, R. and Gerstner, W. (2005). Adaptive exponential integrate-and-fire J. Neurophysiol., 94:3637- 3642.Izhikevich, E. M. (2003). Simple model of spiking neurons. IEEE Trans Neural Netw, 14:1569-1572.Gerstner, W. (2008). Spike-response model. Scholarpedia, 3(12):1343.

Nonlinear and adaptive IF

Optimization methods for neuron models, max likelihood, and GLM

Encoding and Decoding

The END

( ) exp( ) ( )rest

du uu u w RI t

dt

( ) ( )fw rest w f

dwa u u w b t t

dt

u-nullclinea=0

u

What happens if input switches from I=0 to I>0?

u-nullcline[ ] u-nullcline moves horizontally[ ] u-nullcline moves vertically[ ] w-nullcine moves horizontally[ ] w-nullcline moves vertically

Only during reset

Neuronal Dynamics – Quiz 9.2. Nullclines for constant input

( )n k n k restk

u t k I u

2

1

[ ( ) ]K

datan k n k rest

n k

E u t k I u

( )n nu t k x

Neuronal Dynamics – 9.5 Parameter estimation: voltage Vector notation

( )n nu t k x

0



Linear in parameters = linear fit = quadratic problem

( )n nu t k x

2

1

[ ( ) ]K

datan k n k rest

n k

E u t k I u

0

( )s S t s ds







- Firing patterns and adaptation9.3 Spike Response Model (SRM) - Integral formulation9.4 Generalized Linear Model - Adding noise to the SRM 9.5 Parameter Estimation - Quadratic and convex optimization9.6. Modeling in vitro data - how long lasts the effect of a spike?9.7. Helping Humans

Week 9 – part 7: Helping Humans

-Predict spike times-Predict subthreshold voltage-Easy to interpret (not a ‘black box’)-Variety of phenomena-Systematic: ‘optimize’ parameters

Neuronal Dynamics – Review: Models and Data

BUT so far limited to in vitro

Now: extracellular recordings

visual cortex

Model of ‘Encoding’

A) Predict spike times, given stimulusB) Predict subthreshold voltageC) Easy to interpret (not a ‘black box’)D) Flexible enough to account for a variety of phenomenaE) Systematic procedure to ‘optimize’ parameters

Neuronal Dynamics – 7.7 Systems neuroscience, in vivo

Estimation of spatial (and temporal) receptive fields

( ) k K k restu t k I u firing intensity ( ) ( ( ) ( ))t f u t t

LNP

Neuronal Dynamics – 7.7 Estimation of receptive fields

Special case ofGLM=Generalized Linear Model

LNP = Linear-Nonlinear-Poisson

visualstimulus

Neuronal Dynamics – 7.7 Estimation of Receptive Fields

GLM for prediction of retinal ganglion ON cell activity

Pillow et al. 2008

Neuronal Dynamics – 7.7 GLM with lateral coupling

Pillow et al. 2008

One cell in a Network of Ganglion cells

visual cortex


A) Predict spike times, given stimulusB) Predict subthreshold voltageC) Easy to interpret (not a ‘black box’)D) Flexible enough to account for a variety of phenomenaE) Systematic procedure to ‘optimize’ parameters

Neuronal Dynamics – 7.7 Model of ENCODING


Neuronal Dynamics – 7.7 ENCODING and Decoding

Generalized Linear Model (GLM) - flexible model - systematic optimization of parameters

Model of ‘Decoding’The same GLM works! - flexible model - systematic optimization of parameters

visual cortex

Model of ‘Decoding’: predict stimulus, given spike times

Neuronal Dynamics – 7.7 Model of DECODING

Predict stimulus!

Model of ‘Decoding’

Predict intended arm movement, given Spike Times

Application: Neuroprostheticsfrontal cortex

Neuronal Dynamics – 7.7 Helping Humans

Many groupsworld wide work on this problem!

motor cortex

Application: Neuroprosthetics

Hand velocity

Decode the intended arm movement

Neuronal Dynamics – 7.7 Basic neuroprosthetics

Mathematical models for neuroscience

help humans

The end

Neuronal Dynamics – 7.7 Why mathematical models?

biological modeling of neural networks:

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