Stable Propagation of Synchronous Spiking in Cortical Neural ... - FIAS

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.