Biological Modeling of Neural Networks: Week 9 – Optimizing Neuron Models For Coding and Decoding Wulfram Gerstner EPFL, Lausanne, Switzerland 9.1 What.

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Biological Modeling of Neural Networks: Week 9 – Optimizing Neuron Models For Coding and Decoding Wulfram Gerstner EPFL, Lausanne, Switzerland 9.1 What is a good neuron model? - Models and data 9.2 AdEx model - Firing patterns and adaptation 9.3 Spike Response Model (SRM) - Integral formulation 9.4 Generalized Linear Model - Adding noise to the SRM 9.5 Parameter Estimation - Quadratic and convex optimization 9.6. 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 times B)Predict subthreshold voltage C)Easy to interpret (not a ‘black box’) D)Flexible enough to account for a variety of phenomena E)Systematic procedure to ‘optimize’ parameters Neuronal Dynamics – 9.1 What is a good neuron model?

What is a good choice of f ? Neuronal Dynamics – Review : Nonlinear Integrate-and-fire See: week 1, lecture 1.5

What is a good choice of f ? (i) Extract f from more complex models (ii) Extract f from data (2) (1) Neuronal Dynamics – Review : Nonlinear Integrate-and-fire

(i) Extract f from more complex models See week 3: 2dim version of Hodgkin-Huxley Separation of time scales: Arrows are nearly horizontal resting state Spike initiation, from rest A. detect spike and reset B. Assume w=w rest Neuronal Dynamics – Review : Nonlinear Integrate-and-fire

(i) Extract f from more complex models Separation of time scales linear exponential See week 4: 2dim version of Hodgkin-Huxley Neuronal Dynamics – Review : Nonlinear Integrate-and-fire

(ii) Extract f from data Pyramidal neuron Inhibitory interneuron linear exponential Badel et al. (2008) Exp. Integrate-and-Fire, Fourcaud et al Neuronal Dynamics – Review : Nonlinear Integrate-and-fire linear exponential

Best choice of f : linear + exponential (1) (2) BUT: Limitations – need to add -Adaptation on slower time scales -Possibility for a diversity of firing patterns -Increased threshold after each spike -Noise Neuronal Dynamics – Review : Nonlinear Integrate-and-fire

Biological Modeling of Neural Networks: Week 9 – Optimizing Neuron Models For Coding and Decoding Wulfram Gerstner EPFL, Lausanne, Switzerland 9.1 What is a good neuron model? - Models and data 9.2 AdEx model - Firing patterns and adaptation 9.3 Spike Response Model (SRM) - Integral formulation 9.4 Generalized Linear Model - Adding noise to the SRM 9.5 Parameter Estimation - Quadratic and convex optimization 9.6. Modeling in vitro data - how long lasts the effect of a spike? 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

Add adaptation variables: jumps by an amount after each spike SPIKE AND RESET 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)

AdEx model Can we understand the different firing patterns? Phase plane analysis! Neuronal Dynamics – 9.2 Adaptive Exponential I&F

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 minute Restart at 9:38

Biological Modeling of Neural Networks: Week 9 – Optimizing Neuron Models For Coding and Decoding Wulfram Gerstner EPFL, Lausanne, Switzerland 9.1 What is a good neuron model? - Models and data 9.2 AdEx model - Firing patterns and adaptation 9.3 Spike Response Model (SRM) - Integral formulation 9.4 Generalized Linear Model - Adding noise to the SRM 9.5 Parameter Estimation - Quadratic and convex optimization 9.6. Modeling in vitro data - how long lasts the effect of a spike? Week 9 – part 2b : Firing Patterns

AdEx model w jumps by an amount b after each spike u is reset to u r after each spike parameter a – slope of w -nullcline Can we understand the different firing patterns?

AdEx model – phase plane analysis: large b b u-nullcline u is reset to u r a=0

AdEx model – phase plane analysis: small b b u-nullcline u is reset to u r adaptation

Quiz 9.2: AdEx model – phase plane analysis b u-nullcline u is reset to u r What firing pattern do you expect? (i) Adapting (ii) Bursting (iii) Initial burst (iv)Non-adapting

b u-nullcline u is reset to u r AdEx model – phase plane analysis: a >0

Firing patterns arise from different parameters! w jumps by an amount b after each spike u is reset to u r after 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

Best choice of f : linear + exponential (1) (2) BUT: Limitations – need to add -Adaptation on slower time scales -Possibility for a diversity of firing patterns -Increased threshold after each spike -Noise Neuronal Dynamics – Review : Nonlinear Integrate-and-fire

Add dynamic threshold: Threshold increases after each spike Neuronal Dynamics – 9.2 AdEx with dynamic threshold

add -Adaptation variables -Possibility for firing patterns -Dynamic threshold -Noise Neuronal Dynamics – 9.2 Generalized Integrate-and-fire

Biological Modeling of Neural Networks: Week 9 – Optimizing Neuron Models For Coding and Decoding Wulfram Gerstner EPFL, Lausanne, Switzerland 9.1 What is a good neuron model? - Models and data 9.2 AdEx model - Firing patterns and adaptation 9.3 Spike Response Model (SRM) - Integral formulation 9.4 Generalized Linear Model - Adding noise to the SRM 9.5 Parameter Estimation - Quadratic and convex optimization 9.6. Modeling in vitro data - how long lasts the effect of a spike? Week 9 – part 3: Spike Response Model (SRM)

Exponential versus Leaky Integrate-and-Fire Reset if u= Badel et al (2008) Leaky Integrate-and-Fire

jumps by an amount after each spike SPIKE AND RESET Dynamic threshold Neuronal Dynamics – 9.3 Adaptive leaky integrate-and-fire

Linear equation  can be integrated! Spike Response Model (SRM) Gerstner et al. (1996) Neuronal Dynamics – 9.3 Adaptive leaky I&F and SRM Adaptive leaky I&F

Spike emission potential threshold Arbitrary Linear filters i Input I(t) Gerstner et al., 1993, 1996 u(t) Neuronal Dynamics – 9.3 Spike Response Model (SRM)

SRM with appropriate leads to bursting Neuronal Dynamics – 9.3 Bursting in the SRM

Exercise 1: from adaptive IF to SRM Integrate the above system of two differential equations so as to rewrite the equations as potential A – what is ? B – what is ? (i)(ii) (iii) (iv) Combi of (i) + (iii) Next lecture at 9:57/10:15

potential threshold Input I(t) S(t) + firing if Gerstner et al., 1993, 1996 Neuronal Dynamics – 9.3 Spike Response Model (SRM)

+ Linear filters for - input - threshold - refractoriness potential threshold

Biological Modeling of Neural Networks: Week 9 – Optimizing Neuron Models For Coding and Decoding Wulfram Gerstner EPFL, Lausanne, Switzerland 9.1 What is a good neuron model? - Models and data 9.2 AdEx model - Firing patterns and adaptation 9.3 Spike Response Model (SRM) - Integral formulation 9.4 Generalized Linear Model - Adding noise to the SRM 9.5 Parameter Estimation - Quadratic and convex optimization 9.6. Modeling in vitro data - how long lasts the effect of a spike? Week 9 – part 4: Generalized Linear Model (GLM)

Spike Response Model (SRM) Generalized Linear Model GLM potential threshold firing intensity Gerstner et al., 1992,2000 Truccolo et al., 2005 Pillow et al I(t) S(t) +

escape process u(t) t escape rate u Neuronal Dynamics – review from week 8: Escape noise escape rate Example: leaky integrate-and-fire model

Exerc. 2.1: Non-leaky IF with escape rates Integrate for constant input (repetitive firing) Next lecture at 10:38 Calculate - potential - hazard - survivor function - interval distrib. nonleaky

escape process u(t) Survivor function t escape rate Interval distribution Survivor function escape rate u Good choice Neuronal Dynamics – review from week 8: Escape noise

Measured spike train with spike times Likelihood L that this spike train could have been generated by model? Neuronal Dynamics – 9.4 Likelihood of a spike train in GLMs  Blackboard

Neuronal Dynamics – 9.4 Likelihood of a spike train

+ Neuronal Dynamics – 9.4 SRM with escape noise = GLM -linear filters -escape rate  likelihood of observed spike train  parameter optimization of neuron model

Biological Modeling of Neural Networks: Week 9 – Optimizing Neuron Models For Coding and Decoding Wulfram Gerstner EPFL, Lausanne, Switzerland 9.1 What is a good neuron model? - Models and data 9.2 AdEx model - Firing patterns and adaptation 9.3 Spike Response Model (SRM) - Integral formulation 9.4 Generalized Linear Model - Adding noise to the SRM 9.5 Parameter Estimation - Quadratic and convex optimization 9.6. Modeling in vitro data - how long lasts the effect of a spike? Week 9 – part 5: Parameter Estimation

Subthreshold potential Linear filters/linear in parameters known spike trainknown input Neuronal Dynamics – 9.5 Parameter estimation: voltage I(t) S(t) + Spike Response Model (SRM) Generalized Lin. Model (GLM)

Linear in parameters = linear fit = quadratic problem Neuronal Dynamics – 9.5 Parameter estimation: voltage comparison model-data Blackboard: Riemann-sum

Linear in parameters = linear fit Neuronal Dynamics – 9.5 Parameter estimation: voltage Blackboard: Error function I

Model Data I Linear in parameters = linear fit = quadratic optimization Neuronal Dynamics – 9.5 Parameter estimation: voltage

Mensi et al., 2012 Subthreshold potential known spike train known input pyramidal inhibitory interneuron pyramidal Neuronal Dynamics – 9.5 Extracted parameters: voltage

Exercise 3 NOW: optimize 1 free parameter Model Data I Optimize parameter R, so as to have a minimal error

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?

Biological Modeling of Neural Networks: Week 9 – Optimizing Neuron Models For Coding and Decoding Wulfram Gerstner EPFL, Lausanne, Switzerland 9.1 What is a good neuron model? - Models and data 9.2 AdEx model - Firing patterns and adaptation 9.3 Spike Response Model (SRM) - Integral formulation 9.4 Generalized Linear Model - Adding noise to the SRM 9.5 Parameter Estimation - Quadratic and convex optimization 9.6. Modeling in vitro data - how long lasts the effect of a spike? Week 9 – part 5b: Quadratic and Convex Optimization

Fitting models to data: so far ‘subthreshold’ Adaptation current Dyn. threshold

potential threshold firing intensity I(t) S(t) + Jolivet&Gerstner, 2005 Paninski et al., 2004 Pillow et al Neuronal Dynamics – 9.5 Threshold: Predicting spike times

potential threshold firing intensity Neuronal Dynamics – 9.5 Generalized Linear Model (GLM)

potential threshold firing intensity 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

potential threshold I(t) S(t) + firing intensity Quiz NOW : What are the units of ? (i)Voltage? (ii)Resistance? (iii)Resistance/s? (iv)Current?

potential threshold I(t) S(t) + firing intensity Quiz NOW: What are the units of ? (i)Voltage? (ii)Resistance? (iii)Resistance/s? (iv)Current?

Biological Modeling of Neural Networks: Week 9 – Optimizing Neuron Models For Coding and Decoding Wulfram Gerstner EPFL, Lausanne, Switzerland 9.1 What is a good neuron model? - Models and data 9.2 AdEx model - Firing patterns and adaptation 9.3 Spike Response Model (SRM) - Integral formulation 9.4 Generalized Linear Model - Adding noise to the SRM 9.5 Parameter Estimation - Quadratic and convex optimization 9.6. 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

potential threshold firing intensity I(t) S(t) + Jolivet&Gerstner, 2005 Paninski et al., 2004 Pillow et al 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

potential threshold firing intensity Change in model formulation: What are the units of …. ? I(t) S(t) + exponential adaptation current ‘soft-threshold adaptive IF model’

Pozzorini et al Time scale of filters?  Power law A single spike has a measurable effect more than 10 seconds later! 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 references Reading: 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, Brillinger, D. R. (1988). Maximum likelihood analysis of spike trains of interacting nerve cells. Biol. Cybern., 59: 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: Paninski, L. (2004). Maximum likelihood estimation of … Network: Computation in Neural Systems, 15: Paninski, L., Pillow, J., and Lewi, J. (2007). Statistical models for neural encoding, decoding, and optimal stimulus 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: 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: Mensi, S., et al. (2012). Parameter extraction and classication …. J. Neurophys.,107: 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: Donoghue, J. (2002). Connecting cortex to machines: recent advances in brain interfaces. Nat. Neurosci., 5: Fourcaud-Trocme, N., Hansel, D., van Vreeswijk, C., and Brunel, N. (2003). How spike …. J. Neuroscience, 23: Badel, L., et al. (2008a). Extracting nonlinear integrate-and-fire, Biol. Cybernetics, 99: Brette, R. and Gerstner, W. (2005). Adaptive exponential integrate-and-fire J. Neurophysiol., 94: Izhikevich, E. M. (2003). Simple model of spiking neurons. IEEE Trans Neural Netw, 14: 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

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u-nullcline a=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

Neuronal Dynamics – 9.5 Parameter estimation: voltage Vector notation

Linear in parameters = linear fit = quadratic problem Neuronal Dynamics – 9.5 Parameter estimation: voltage

Biological Modeling of Neural Networks: Week 9 – Optimizing Neuron Models For Coding and Decoding Wulfram Gerstner EPFL, Lausanne, Switzerland 9.1 What is a good neuron model? - Models and data 9.2 AdEx model - Firing patterns and adaptation 9.3 Spike Response Model (SRM) - Integral formulation 9.4 Generalized Linear Model - Adding noise to the SRM 9.5 Parameter Estimation - Quadratic and convex optimization 9.6. 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 stimulus B)Predict subthreshold voltage C)Easy to interpret (not a ‘black box’) D)Flexible enough to account for a variety of phenomena E)Systematic procedure to ‘optimize’ parameters Neuronal Dynamics – 7.7 Systems neuroscience, in vivo

Estimation of spatial (and temporal) r eceptive fields firing intensity LNP Neuronal Dynamics – 7.7 Estimation of receptive fields

Special case of GLM= Generalized Linear Model LNP = Linear-Nonlinear-Poisson visual stimulus 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 One cell in a Network of Ganglion cells

visual cortex Model of ‘Encoding’ A)Predict spike times, given stimulus B)Predict subthreshold voltage C)Easy to interpret (not a ‘black box’) D)Flexible enough to account for a variety of phenomena E)Systematic procedure to ‘optimize’ parameters Neuronal Dynamics – 7.7 Model of ENCODING

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: Neuroprosthetics frontal cortex Neuronal Dynamics – 7.7 Helping Humans Many groups world 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?