Synaptic plasticity : The linear superposition approach

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Synaptic plasticity The linear superposition
approach
Song, Miller Abbott (2000)
Start with
mathematically
Note I often use an inverted notation
Balanced Network
Here use t t-20 ms A-/A1.05
1000 x 200 x
Excitatory inhibitory
Excitatory rates vary Inhibitory rate 10 Hz
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• How is this actually done.
• The postsynaptic neuron is an integrate and fire
  neuron.
• Presynaptic activity is generated randomly
  Poisson spikes, or correlated poisson.
• How are synaptic weight changes actually
  generated?

Given a presynaptic AP
otherwise Same hold for
inhibitory synapses. When VVth an AP is pasted
and the neuron is reset to Vrest
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How is synaptic plasticity actually
implemented. -all spike pairs count N1
functions Pa and M. Every time there is a post
spike M ?M-A- every time there is a pre spike
on neuron (a) Pa?PaA. Otherwise If there
is a postspike - ga?gaPa(t)gmax while
galtgmax If there is a postspike - ga?gaM(t)gmax
while galtgmax Synapses are bounded between gmin
and gmax
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Describe random input simulations on board to
show how this scheme accomplishes what wee expect
from all to all STDP
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What are the consequences of this setup?
40 Hz
10 Hz
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Why does this happen?
A
Before learning
Alt A-
A-
After learning
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Effect of presynaptic correlations
Low correlation ? High correlation
t20 ms
t200 ms
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• Homework 5 Plasticity in one synapse with the
  Song et. al model (due 4/9).
• In this homework there will be no postsynaptic
  model neuron. Instead we will simulate only
  conditions in which we externally determine the
  post spikes.
• Code a plasticity rule as in Song. et al. 2000,
  using M and P as above. For this synapse,
  generate 100 spike pairs with different ?t from
  -7070. Show the time evolution of the synaptic
  conductance for ?t10 and ?t-10. Show the
  complete resulting STDP curve from -70 to 70
  (with jumps of 5).
• Change t- to 40 and cut A- in half. Show the
  resulting STDP curve.

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c. Generate random pre and postsynaptic spikes at
10 Hz. Show the time evolution of the weights
for A-/A1.05 and for A-/A0.95. How does the
rate of the pre and postspikes effect the final
conductances (weights). d. Add a small
correlation between pre and post spikes and see
how this correlation effects the development of
weights. One way to do it is to add to the
random postspikes additional postspikes which
come a fixed time after some of the prespikes.
For example, randomly for every 10th prespike
add a postspike with a 1 ms latency. Do this at
A-/A1.05 and show how changing the correlation
affects the weight evolution.
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• Some key questions
• How can we analyze this theory ?
• Are its consequences consistent with
  experimental results ?

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Analysis of a similar model (Kempter, Gerstner,
van Hemmen. 1999) How is the STDP model related
to firing rate models?
Input Output
Plasticity (change from time t to tt)
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denote
t-t?ts
Averaging over a random processes lt gt Get
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If t is large enough, such that many pre and post
spikes occur in time t, are self averaging, we
can write Use ltxigt?ini, ltygt ?out. If most
of the mass of F is in a range much smaller than
t we can extend the integral for ds from -8 to 8.
Get
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From Song et al. paper
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The correlation between input and output
is For the linear neuron, in the rate based
case we could rewrite this as a function of the
inputs, and input correlations alone. Something
similar can be done here. Assume pre and
postsynaptic spikes are non-homogeneous Poisson
(ah?)
Use linear model for rate
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math (Read paper online) Get
Where
etc
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Similar to the Linsker equation

• short time correlations matter
• If mean of F(s)lt0 k2 might be smaller than 0

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This is really beautiful, but 1. Does this
linear superposition theory agree with
experimental results in terms of different
induction paradigms? 2. Do consequences of a
Linsker-like model agree with experimental
results in terms of RF plasticity?
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• Rate based synaptic plasticity cannot be
  accounted for by superposition

10 Hz
40 Hz
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2. Even STDP is rate dependent
Sojtstrom et. al 2001
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3. LTD at 1 Hz does not generally generate any
post spikes (I am not sure about this)
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4. STDP induced by triplets and quadruplets
cannot be accounted for by superposition of pairs
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Another line of evidence has to do with formation
of receptive fields. The logic here is more
complex. Previously (class 2) we studied the
Hebb and the Oja (PCA) rule. We find that these
rules do not account well for many properties of
recepeptive fields. According to the analysis
above (Kempter et. al.) the STDP rule is formally
very similar to the Linsker rule, and therefore
is likely to exhibit the same problems.