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Stable Propagation of Synchronous Spiking in
Cortical Neural Networks
Markus Diesmann, Marc-Oliver Gewaltig, Ad AertsenNature 402:529-533
Flavio FrohlichComputational 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)NeuralSystem 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 VSteady state membrane voltage EL
Input resistance Rm
Transmembrane current IE
• 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?
susta
ined
dies out
Input to Model Neuron
• Pulse packet: spike volley.– Activity a: number
of spikes in volley.– Temporal dispersion
: standard deviation of underlying pulse time distribution.
-3 -2 -1 0 1 2 30
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
in
-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.1
0.2
0.3
0.4
0.5 a = 20
Pulse packet
Output = Neuron(Input)
• Input to model neurons: pulse packets (pooling from many neurons in previous layer).
-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.1
0.2
0.3
0.4
0.5
I&F Neuron-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.1
0.2
0.3
0.4
0.5
-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.1
0.2
0.3
0.4
0.5
I&F Neuron
I&F Neuron
-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.1
0.2
0.3
0.4
0.5
-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.1
0.2
0.3
0.4
0.5
-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.1
0.2
0.3
0.4
0.5
• Output of model neuron: at most one spike.
• Spike probability • Standard deviation
out.
Neural Transmission Function I
Inputdispersion in
# input spikes
Sp
ikin
g p
rob
abili
ty
Neural Transmission Function II
Input dispersion
# input spikes
Ou
tpu
t d
isp
ersi
on
in>out
out>in
-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
in
-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
out
-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
in
-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
out
State Space Analysis
Stable attractor
Saddle point
State-space analysis of propagating spike synchrony.
State variables:Activity aDispersion
Trajectory t=t(i) where i denotes ordered group.
Size of Neuron Groups WW = 80 W = 90 W = 100 W = 110
zero-isocline activity a
zero-isocline dispersion
region of attraction
• Increase W Fixpointsmove 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.