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.