Neuroinformatics

April 6, 2017

Lecture 6: Synaptic plasticity and Hebb’s rule

Learning what is is?

Learning what is is?

Action-perception loop

Types of plasticity

I Structural plasticity is the mechanism describing thegeneration of new connections and thereby redefining thetopology of the network.

I Functional plasticity is the mechanism of changing the strengthvalues of existing connections.

Hebbian plasticity

”When an axon of a cell A is near enough to excitecell B or repeatedly or persistently takes part in firing it,some growth or metabolic change takes place in bothcells such that A’s efficiency, as one of the cells firing B,is increased.”

Donald O. Hebb, The organization of behavior, 1949See also Sigmund Freud, Law of association by simultaneity, 1888Santiago Ramn y Cajal - memories might instead be formed bystrengthening the connections between existing neurons to improvethe effectiveness of their communication, 1894

Possible neuronal mechanisms sub-serving learning and memory

Synaptic mechanism

Long term potentiaition

Hebbian model

Association

r ini

wi r out

Neuron model: In each time step the model neurons fires if∑i wi r in

i > 1.5Learning rule: Increase the strength of the synapses by a value∆w = 0.1 if a presynaptic firing is paired with a postsynaptic firing.

Associative learning

A.

r in

r = 1out

1010100000

w1010100000

r in

r = 0out

0000000111

w1010100000

B.

r in

r = 1out

1010100111

w1.101.101.1000.10.10.1

C.

r in

0000000111

r = 1out

w1.601.601.6000.60.60.6

D.Before learning,only adour cue

Before learning,only visual cue

After 1 learning step, both cues

After 6 learning steps,only visual cue

Features of associators and Hebbian learning

I Pattern completion and generalization, recall from partial input,overlap between input and trained pattern (recognition of noisynumbers)

I Prototypes and extraction of central tendencies, training on manysimilar but not equivalent examples (individual face, manycommon features in all faces)

I Graceful degradation and fault tolerance (loss of synapses orwhole neurons)

Hippocampus

I Hippocampus: centre of memory storage, The dentate gyrus isthought to contribute to the formation of new memories. It isnotable as being one of a select few brainstructures currentlyknown to have high rates of neurogenesis in adult rats

I Neurons must be plasticI Experiment: isolated slices of hippocampal tissue placed in

dishes

LTP experimentI EXPERIMENTAL confirmation of Hebb’s rule (1949)I i) single pulse is presented ii) stimulation with burst of pulses:

100 pulses/sec ii) After LTP induced, single pulse stimulationI Postsynaptic cells must be depolarized to LTP be produced AND

receiving excitatory input - see Associative learning slide.

Original LTP by Bliss and Lomo, 1973

I Long-lasting changes of synaptic response characteristicsI High frequency-stimulus is applied (plasticity-induced tetanus)→

long-term potentiation(to strengthen, make more potent) (LTP)average amplitude of EPSP increased

I Long frequency stimulus→ long-term depression (LTD)

Classical LTP and LTD

A. Long term potentiation

100

140

80

160

180

120

-10 10 20 300

Cha

nge

in E

FP a

mpl

itude

[%]

Time [min]

-10 10 20 300

100

120

140

60

80

40

Time [min]Cha

nge

in E

FP a

mpl

itude

[%]

B. Long term depression

Spike timing dependent plasticity (STDP)I Bi-Poo experiments: voltage clamp for hippocamal cells in vitro,→ Excitatory PostSynaptic Current (EPSP)→ critical timewindow ∆t = 40ms

I critical window width is much larger, asymmetrical andsymmetrical (for bursting neurons) form of Hebbian plasticity,inverse correlation in Purkinje cells (inhibitory) in the cerebellum

LTP

LTD

D.

B. LTP

LTDt − tpost pre

t − tpost pre

LTP

LTD

t − tpost pre

0

Δw

0

Δw

0

Δw

C. LTP

LTDt − tpost pre

0

Δw

E. LTP

LTD

t − tpost pre

0

Δw

t − t −80

−80−60−20

0204060

80100

80400−40

Cha

nge

in E

PSC

am

plitu

de [%

]

post pre

A. Spike timing dependent plasticity

[ms]

Morris Water Maze - spatial memory

I i) mice training ii) Chemical blocking of LTP by AP5 impair spatiallearning, keep control group iii) AP5-treated mice significantlyimpaired

I i) slices of the hippocampus were taken from both groups ii) LTPwas easily induced in controls, but could not be induced in thebrains of APV-treated rats

I Alzheimer’s disease→ cognitive decline seen in individuals withAD may result from impaired LTP ??

Mathematical formulation of Hebbian plasticity - spiking models

wij (t + ∆t) = wij (t) + ∆wij (t fi , t

fj ,∆t ; wij ).

∆w±ij = ε±(w)K±(tpost − tpre)

Spike Timing Dependent Plasticity (SPDP)¿ (i) Exponential plasticitycurve, (ii) Repeated spike pairings induced w UNBOUNDED growth→ a weight dependent learning rate ε±

∆w±ij = ε±(w)e∓

tpost−tpre

τ± Θ(±[tpost − tpre]).

Additive rule with hard (absorbing) boundaries:

ε± =

{a± for wmin

ij ≤ wij ≤ wmaxij

0 otherwise,

Multiplicative rule (soft boundaries):

ε+ = a+(wmax − wij )

ε− = a−(wij − wmin). (1)

The LIF-neuron noise simulation I

I real neuron with 5000 presynaptic neuronI 10 % simulation→ 500 Poisson-distributed spike trains (??) with

refractory correctionsI mean firing rate = 20 Hz, after correction 19.3 Hz, refractory

constant 2 ms.I each presynaptic spike→ EPSP in form of α function (??)I ω = 0.5→ regular firing, CV = 0.12, average rate 118 Hz.I ω = 0.25→ irregular firing, CV = 0.58, average rate 16 Hz. The

CV > lower bound found in experiments

The LIF-neuron noise simulation II

0 200 400 600 800 10000

5

10

15

20

25

30

Time [ms]

RI e

xt

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

P(IS

I)

C = 0.12V

0 50 100 150 200 250 3000

0.05

0.1

0.15

0.2

ISI [ms]

C = 0.58V

A. Time varying input

B. Normalized histogram of ISI C. Normalized histogram of ISI

Threshold

ISI [ms]

P(IS

I)

Synaptic scaling and weight distributionsI IF neuron with 1000 excitatory synapses driven by presynaptic

Poisson spike trains with average firing rate of 20 Hz,∆w±

ij = ε±(w)K±(tpost − tpre) applying additive rule andasymmetrical Gaussian plasticity windows

I (i) weights set to large values (ii) large frequency firing (see lec4)(iii) apply additive STDP rule with marginally stronger LTD thanLTP

I increased CV, firing rate reduction, weight BINOMICALdistribution after 5 mins

0 0.5 1 1.5 2 2.5 30

50

100

150

200

250

Time [min]

Firin

g ra

te [H

z]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

C

0 0.005 0.01 0.0150

50

100

150

200

250

Weight values

Num

ber

A. Learning phase B. Weight distribution after learning

VCV

Firing rate

after Song, Miller and Abbott 2000

Cross-correlation functionI s(∆t), s = 1 if a spike occurs in ∆tI star line:C(n) = 0 for regular IF firing 270 Hz, w = 0.015, LTP

occurs as much as LTDI square line:after Hebb’s learning, IF firing 18 Hz, some

presynaptic spikes elicits post-synaptic spikesI C < 0, if presynaptic spikes reduce postsynaptic

(anti-correlation) and vice-versa

C(n) = 〈spre(t)spost (t + nδt)〉 − 〈sprespost〉

−20 −10 0 10 20−0.05

0

0.05

0.1

0.15

0.2

Δ t [ms]

Aver

age

cros

s-co

rrela

tion

after Song, Miller and Abbott 2000