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What Can a Neuron Learn with Spike-Timing-Dependent Plasticity?
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STDP Finds the Start of Repeating Patterns in Continuous Spike Trains
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Spiking neurons are flexible computational
modules.
Enable implement with their adjustable synaptic
parameters an enormous variety of different
transformations from input spike trains to output
spike trains.
The perceptron convergence theorem asserts the
convergence of a supervised learning algorithm.
In contrast, no guarantee for the convergence of
STDP with teacher forcing that holds for arbitrary
input spike patterns.
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On the other hand, hold for STDP in the case
of Poisson input spike trains.
The resulting necessary and sufficient
condition can be formulated in terms of
linear separability.
◦ In case of perceptrons (McCulloch-Pitts neurons):
threshold gates with static synapses, static batch
inputs and outputs.
◦ In case of STDP: time-varying input and output
streams.
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The theoretically predicted convergence of
STDP with teacher forcing also holds for more
realistic neurons models, dynamic synapses,
and more general input distributions.
The positive learning results hold for different
interpretations of STDP where:
◦ changes the weights of synapses
◦ modulates the initial release probability of dynamic
synapses
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STDP is related to various important learning
rules and learning mechanisms.
Question: To what extent STDP might support a
more universal type of learning where a neuron
learns to implement an “arbitrary given” map?
There exist many maps from input spike trains to
output spike trains that can’t be realized by a
neuron for any setting of its adjustable
parameters.
◦ For example, no values of weight could enable a generic neuron to
produce a high-rate output spike train in the absence of any input
spikes.
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A neuron learn to implement transformations
in a stable manner with a parameter setting
that represents a equilibrium point for the
learning rule under consideration (STDP).
STDP always produces bimodal distribution
of weights,
the minimal or
maximal possible.
◦ Need to consider such conditions.
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Which of the many parameters that influence the
input-output behavior should be viewed as being
adjustable for a specific protocol for inducing
synaptic plasticity (i.e., “learning”)?
STDP adjust the following parameters:
◦ scaling factors w of the amplitudes
◦ initial release probabilities U
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Whereas:
◦ An increase of parameter U will increase the amplitude
of the EPSP for the first spike.
◦ An increase of the scaling factor w tends to decrease the
amplitudes of shortly following EPSPs.
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Assumption: during learning, the neuron is taught to
fire at particular points in time via extra input
currents,
◦ which could represent synaptic inputs from other
cortical or subcortical areas.
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SNCC – spiking neuron convergence conjecture:
STDP enables neurons to learn under this protocol:
◦ starting with arbitrary initial values
◦ any input-output transformation that the neuron could
implement
◦ in a stable manner for some values of its adjustable
parameters.
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A standard leaky integrate-and-fire neuron model:
dVm
m
 (Vm  Vresting )  Rm  ( I syn (t )  I background  I inject (t ))
dt
◦ Vm = membrane potential
◦  m = membrane time constant
◦ Rm = membrane resistance
◦ I syn (t ) = the current supplied by the synapse
◦ I background = a constant background current
◦ I inject (t ) = currents induced by a “teacher”
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If Vm exceeds the threshold voltage, it is reset
and held there for the length of the absolute
refractory period.
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The model proposed in Markram, Wang and
Tsodyks (1998), predicts the amplitude of the
excitatory postsynaptic current (EPSC) for the
kth spike in a spike train with interspike
intervals 1, 2 ,..., k 1 through the equations:
Ak  w  uk  Rk
uk  U  uk 1 (1  U )e

 k 1
F
Rk  1  ( Rk 1  uk 1 Rk 1  1)e

 k 1
D
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The variables u  [0,1], R  [0,1] are dynamic, whose
initial values for the first spike are u1  U , R1  1
The parameters U, D, and F were randomly chosen
from gaussian distributions that were based on
empirically found data for such connections:
◦ If the input was excitatory (E) the mean values of these
three parameters (with D, F expressed in seconds) were
chosen to be 0.5, 1.1, 0.05.
◦ If the input was inhibitory (I) then 0.25, 0.7, 0.02.
◦ The SD of each parameter was chosen to be 10% of its
mean.
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The effect of STDP is commonly tested by measuring
in the postsynaptic neuron the amplitude A1 of the
EPSP for a single spike from the presynaptic neuron.
The interpretations for any change A in the
amplitude of A1  w U  R1 can caused by :
◦ A proportional change w of the parameter w
◦ A proportional change U of the initial release
probability u1 = U
◦ A change of both w and U
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According to Abbott & Nelson (2000), the change A1
in the amplitude A1 of EPSPs (for the first spike in a
test spike train) that results from pairing of:
pre
◦ a firing of the presynaptic neuron at some time t
post
 t pre  t
◦ a firing of the postsynaptic neuron at time t
can be approximated for many cortical synapses by
terms of the form:
 W  e  t /  , if
A(t )  
t /  
, if
 W  e
t  0
t  0
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with constants W+,W−, τ+, τ− > 0
◦ with an extra clause that prevents the amplitude A1 from
growing beyond some maximal value Amax or below 0.