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## Part III: Models of synaptic plasticity

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**Swiss Federal Institute of Technology Lausanne, EPFL**Laboratoryof Computational Neuroscience, LCN, CH 1015 Lausanne Part III: Models of synaptic plasticity BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002 Chapters 10-12**Chapter 10: Hebbian Models**• -Hebb rules • STDP BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002 Chapter 10**Hebbian Learning**pre j i k post When an axon of celljrepeatedly or persistently takes part in firing cell i, then j’s efficiency as one of the cells firing i is increased Hebb, 1949 - local rule - simultaneously active (correlations)**pre j**u i spikes of i Hebbian Learning in experiments (schematic) pre j u no spike of i EPSP i post post**pre j**Both neurons simultaneously active i post pre j no spike of i EPSP i Increased amplitude post Hebbian Learning in experiments (schematic) pre j u no spike of i EPSP i post**Hebbian Learning**item memorized**Hebbian Learning**Recall: Partial info item recalled**Hebbian Learning**pre j i k post When an axon of celljrepeatedly or persistently takes part in firing cell i, then j’s efficiency as one of the cells firing i is increased Hebb, 1949 - local rule - simultaneously active (correlations)**activity (rate)**Hebbian Learning: rate model pre j i k post - local rule - simultaneously active (correlations)**+**0 0 0 - - - + 0 - + 0 Hebbian Learning: rate model pre j i k post on on off off pre post on off on off + - - +**Rate-based Hebbian Learning**pre j i k post - local rule - simultaneously active (correlations) Taylor expansion**Rate-based Hebbian Learning**pre j i post a = a(wij) a(wij) wij**Oja’s rule**Rate-based Hebbian Learning pre j i k post**0**Pre before post Spike-based Hebbian Learning pre j i k post - local rule - simultaneously active (correlations)**Spike-based Hebbian Learning**pre j EPSP i k post 0 Pre before post causal rule ‘neuron j takes part in firing neuron’ Hebb, 1949**Spike-time dependent learning window**pre j i post 0 0 0 Pre before post Temporal contrast filter**Spike-time dependent learning window**pre j i post Zhang et al, 1998 review: Bi and Poo, 2001 Pre before post**Spike-time dependent learning: phenomenol. model**pre j i post 0 Pre before post**spike-based Hebbian Learning**pre j post i**Translation invariance**W(tif-tjk ) Learning window spike-based Hebbian Learning pre j BPAP post i**Detailed models**BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002 Chapter 10**Detailed models of Hebbian learning**pre j post i i at resting potential**NMDA**channel i at resting potential Detailed models of Hebbian learning pre j post i**i at high**potential Detailed models of Hebbian learning pre j BPAP post i NMDA channel : - glutamate binding after presynaptic spike - unblocked after postsynaptic spike elementary correlation detector**a**pre b post w Mechanistic models of Hebbian learning pre j BPAP post i**0**Pre before post sophisticated 2-factor Mechanistic models of Hebbian learning pre j BPAP post i pre 4-factor model Gerstner et al. 1998 Buonomano 2001 post Abarbanel et al. 2002**a**pre b post w Mechanistic models of Hebbian learning pre j BPAP post i 1 pre, 1 post**0**Pre before post Mechanistic models of Hebbian learning pre j Dynamics of NMDA receptor (Senn et al., 2001) BPAP post i**Which kind of model?**Descriptive Models Gerstner et al. 1996 Song et al. 2000 Gütig et al. 2003 Mechanistic Models Senn et al. 2000 Abarbanel et al. 2002 Shouval et al. 2000 Optimal Models Chechik, 2003 Hopfield/Brody, 2004 Dayan/London, 2004**Chapter 11: Learning Equations**• -rate based Hebbian learning • STDP BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002 Chapter 11**Rate-based Hebbian Learning**pre j i post a = a(wij) a(wij) wij**Analysis of rate-based Hebbian Learning**x1 x2 xk xk t Linear model Analysis - separation of time scales, expected evolution Correlations in the input**supress index i**eigenvectors Analysis of rate-based Hebbian Learning x1 x2 xk xk t Linear model Correlations in the input**moves towards data cloud**w Analysis of rate-based Hebbian Learning x1 x2 xk xk t x1**becomes aligned**with principal axis w Analysis of rate-based Hebbian Learning x1 x2 xk xk t x1**spike-based Hebbian Learning**pre j post i**Translation invariance**W(tif-tjk ) Learning window spike-based Hebbian Learning pre j BPAP post i**Analysis - separation of time scales, expected evolution**Average over doubly stochastic process Correlations pre/post Analysis of spike-based Hebbian Learning vjk vj1 Point process vk Linear model**Stable if**Rate stabilization (ii) input covariance (plus extra terms) Average over ensemble of rates Covariance of input Analysis of spike-based Hebbian Learning Rewrite equ. (i) fixed point equation for postsyn. rate**Analysis of spike-based Hebbian Learning**(iii) extra spike-spike correlations pre j spike-spike correlations**Spike-based Hebbian Learning**- picks up spatio-temporal correlations on the time scale of the learning window W(s) - non-trivial spike-spike correlations - rate stabilization yields competition of synapses Synapses grow at the expense of others Neuron stays in sensitive regime**Chapter 12: Plasticity and Coding**BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002 Chapter 12**Learning to be fast: prediction**BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002 Chapter 12**Derivative filter and prediction**pre j Mehta et al. 2000,2002 Song et al. 2000 + -**+**- Derivative filter and prediction pre j Mehta et al. 2000,2002 Song et al. 2000 Postsynaptic firing shifts, becomes earlier**Derivative filter and prediction**pre j Mehta et al. 2000,2002 Song et al. 2000 + - derivative of postsyn. rate Roberts et al. 1999 Rao/Sejnowski, 2001 Seung**Learning spike patterns**BOOK: Spiking Neuron Models, W. Gerstner and W. Kistler Cambridge University Press, 2002 Chapter 12**Spike-based Hebbian Learning**pre j EPSP i k post 0 Pre before post causal rule ‘neuron j takes part in firing neuron’ Hebb, 1949