Fundamentals of Computational Neuroscience 2e December 28, 2009 Chapter 4: Associators and synaptic plasticity Types of plasticity I Structural plasticity is the mechanism describing the generation of new connections and thereby redefining the topology of the network. I Functional plasticity is the mechanism of changing the strength values of existing connections. Hebbian plasticity ”When an axon of a cell A is near enough to excite cell B or repeatedly or persistently takes part in firing it, some growth or metabolic change takes place in both cells such that A’s efficiency, as one of the cells firing B, is increased.” Donald O. Hebb, The organization of behavior, 1949 See also Sigmund Freud, Law of association by simultaneity, 1888 Association in ri wi out r Neuron model: In each time step the model neurons fires if P in i wi ri > 1:5 Learning rule: Increase the strength of the synapses by a value ∆w = 0:1 if a presynaptic firing is paired with a postsynaptic firing. A. Before learning, B. Before learning, C. After 1 learning D. After 6 learning steps, only adour cue only visual cue step, both cues only visual cue in in in in r w r w r w r w 1 1 0 1 1 1.1 0 1.6 0 0 0 0 0 0 0 0 1 1 0 1 1 1.1 0 1.6 0 0 0 0 0 0 0 0 1 1 0 1 1 1.1 0 1.6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0.1 1 0.6 0 0 1 0 1 0.1 1 0.6 0 0 1 0 1 0.1 1 0.6 out out out out r = 1 r = 0 r = 1 r = 1 Features of associators and Hebbian learning I Pattern completion and generalization I Prototypes and extraction of central tendencies I Graceful degradation and fault tolerance Classical LTP and LTD A. Long term potentiation B. Long term depression 140 180 120 160 140 100 80 120 100 60 80 40 -10 0 10 20 30 -10 0 10 20 30 Change in EFP amplitude [%] Change Change in EFP amplitude [%] Change Time [min] Time [min] Synaptic neurotransmitter release probability High-frequency stimulation 3 0 EPSP [mV] 100 Time [ms] 100 0 Fluorescence [%] Fluorescence 100 ∆ Time [ms] 150 100 50 p=0.4 p=0.76 p=0.96 Fluorescence [%] Fluorescence ∆ 0 0 5 45 50 90 95 Time [min] Spike timing dependent plasticity A. Spike timing dependent plasticity 100 LTP 80 Δw 60 40 20 t post − pret 0 LTD −20 0 −60 Change in EPSC amplitude [%] −80 −80 −40 0 40 80 t post − pret [ms] C. B. LTP LTP Δw Δw − t post − pret t post pret LTD LTD 0 0 D. LTP E. LTP Δw Δw t post − pret t post − pret LTD LTD 0 0 The calcium hypothesis and modelling chemical pathways D1R mGluR GluR VDCC Gq Golf PLC AC IP3 AMPA cAMP ER Ca2+ CaMKII PDE Ca LTP Store high PKA CaM [Ca 2+ ] PP2A PP2B NMDA Mg low Positive Feedback LTD Loop Thr-75 CDK5 CK1 VGCC DARPP-32 Ser-137 PP2C Thr-34 PP1 I-1 Mathematical formulation of Hebbian plasticity f f wij (t + ∆t) = wij (t) + ∆wij (ti ; tj ; ∆t; wij ): post pre ± ± ∓ t −t post pre τ± ∆wij = (w)e Θ(±[t − t ]): Additive rule with hard (absorbing) boundaries: a± for w min ≤ w ≤ w max ± = ij ij ij ; 0 otherwise Multiplicative rule (soft boundaries): + + max = a (w − wij ) − − min = a (wij − w ): (1) Hebbian learning in population and rate models General: ∆wij = (t; w)[fpost(ri )fpre(rj ) − f (ri ; rj ; w)] Mnemonic equation (Caianiello): ∆wij = (w)[ri rj − f (w)] Basic Hebb: ∆wij = ri rj Covariance rule: ∆wij = (ri − hri i)(rj − hrj i) BCM M BCM theory: ∆wij = (f (ri ; θ )(rj ) − f (w)) − + pre ABS rule: ∆wij = (fABS(ri ; θ ; θ )sign(rj − θ )) Function used in BCM rule Function used in basic ABS rule Threshold for LTP + θ Threshold for LTD − BCM θ ABS f f Μ θ rate r Calcium concentration [Ca 2+ ] Synaptic scaling and weight distributions A. Learning phase B. Weight distribution after learning 250 0.8 250 0.7 200 200 0.6 150 0.5 150 V CV 0.4 C Number 100 100 0.3 Firing rate [Hz] 0.2 50 Firing rate 50 0.1 0 0 0 0 0.5 1 1.5 2 2.5 3 0 0.005 0.01 0.015 Time [min] Weight values after Song, Miller and Abbott 2000 Hebbian rate rules on random pattern 1 X µ µ wij = p (ri − hri i)(rj − hrj i): Np µ 0.15 0.1 Probability 0.05 0 -10 0 10 Synaptic weight Synaptic scaling and PCA wij Explicit normalization: wij P j wij Basic decay: ∆wij = ri rj − cwij Willshaw rule: ∆wij = (ri − wij )rj 2 Oja rule: ∆wij = ri rj − (ri ) wij 1 0.5 x pre r2 0 w2 −0.5 −1 −1 −0.5 0 0.5 1 pre r1 , w1 Further Readings Laurence F. Abbott and Sacha B. Nelson (2000), Synaptic plasticity: taming the beast, in Nature Neurosci. (suppl.), 3: 1178–83. Alain Artola and Wolf Singer (1993), Long-term depression of excitatory synaptic transmission and its relationship to long-term potentiation, in Trends in Neuroscience 16: 480–487. Mark C. W. van Rossum, Guo-chiang Bi, and Gina G. Turrigiano (2000) Stable Hebbian learning from spike timing-dependent plasticity, in J. Neuroscience 20(23): 8812–21.