Stable Propagation of
Synchronous Spiking in Cortical
Neural Networks
Markus Diesmann, Marc

Oliver Gewaltig, Ad
Aertsen
Nature 402:529

533
Flavio Frohlich
Computational Neurobiology UCSD
La Jolla CA

92093
Outline
•
Background
–
Neural Code
–
Integrate&Fire Neuron
•
Motivation / Research Questions
•
Methods
•
Results
•
Discussion & Conclusions
The Neural Code
Stimulus
s
(
t
)
Neural
System
Neural Response
(
t
)
Stimulus
Neural Response
Coding
Given
To determine
Decoding
To determine
Given
The Neural Code
•
Independent

spike versus correlation
code.
•
Temporal versus rate code.
different
The Neural Code
•
Independent

spike code
–
Time

dependent firing rate
r
(
t
).
–
Probability distribution of spike times can be
computed from
r
(
t
) as inhomogenous Poisson
process.
–
Firing rate r(t)
contains all information about
stimulus.
–
Interspike intervals do not carry information.
The Neural Code
•
Correlation code
–
Correlation between spike times carry
information.
–
e.g. information about stimulus carried by
interspike intervals.
The Neural Code
•
Rate code
–
Assumption: independent

spike hypothesis fulfilled.
–
Firing rate
r
(
t
) “varies slowly with time”.
•
Temporal code
–
Assumption: independent

spike hypothesis fulfilled.
–
Firing rate
r
(
t
) “varies rapidly”.
–
“Information is carried by spike timing on a scale
shorter than fastest time characterizing variations of
stimulus.”
–
Requires precise spike timing
millisecond
precision possible for noisy neurons?
Motivation / Research Questions
•
High temporal accuracy observed
in vivo
(precisely timed action potentials related
to stimuli and behavioral events in awake
behaving monkey,
e.g. Abeles 1993
) and
in vitro
.
•
“Can instances of synchronous spiking be
successful transmitted/propagated by
subsequent group of neurons?”
•
“Under which conditions?”
Integrate & Fire Neuron I
•
No biophysical states (channel dynamics).
•
Integrate transmembrane currents.
•
If threshold reached:
–
Stipulate action potential (AP).
–
Reset membrane voltage below threshold.
Integrate & Fire Neuron II
•
Leaky integrate&fire (i&f) neuron:
Time constant
m
Membrane voltage
V
Steady state membrane voltage
E
L
Input resistance
R
m
Transmembrane current
I
E
•
Postsynaptic currents:

function:
•
Background firing (uncorrelated
stationary Poisson distribution)
Network Topology
•
Feedforward architecture.
•
Group = layer.
Group i
Group i+1
•
Each neuron: 20’000
synaptic inputs (88%
excitatory, 12%
inhibitory).
•
100 neurons/group.
•
10 groups.
Predictions
•
“Neurons that share a large enough pool
of simultaneously discharging input cells
tend to align their action potentials.”
•
“A group of neurons can reproduce its
synchronous input activity and act as the
source of synchronous shared input for
the following group.”
Synchronous spiking sustained or
not?
Input to Model Neuron
•
Pulse packet: spike
volley.
–
Activity
a
: number of
spikes in volley.
–
Temporal dispersion
: standard deviation
of underlying pulse
time distribution.
in
a = 20
Pulse packet
Output = Neuron(Input)
•
Input to model neurons:
pulse packets (pooling
from many neurons in
previous layer).
I&F Neuron
I&F Neuron
I&F Neuron
•
Output of model neuron:
at most one spike.
•
Spike probability
•
Standard deviation
out
.
Neural Transmission Function I
Input
dispersion
in
# input spikes
Spiking probability
Neural Transmission Function II
Input dispersion
# input spikes
Output dispersion
in
>
out
out
>
in
State Space Analysis
Stable attractor
Saddle point
State

space analysis
of propagating
spike synchrony.
State variables:
Activity
a
Dispersion
Trajectory
t
=
t
(
i
)
where
i
denotes
ordered group.
Size of Neuron Groups W
W = 80
W = 90
W = 100
W = 110
zero

isocline activity
a
zero

isocline dispersion
region of attraction
•
Increase W
Fixpoints
move apart.
•
Decrease W
Fixpoints
merge to saddle point.
•
Minimal group size W for
maintaining synchrony.
Discussion & Conclusions
•
Stable fixpoint
= 0.5 ms
temporal
precision matching cortical recordings.
•
Region of attraction guarantees
robustness.
•
Model parameters in congruence with
physiological data.
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