Self-Regulated Complexity of Bio-Networks Activity

Eshel Ben Jacob

Eyal Hulata

Itay Baruchi

Ronen Segev

Yoash Shapira

Phys Rev Lett in press

Complexity is still

a blurred intuitive notion

with no agreed upon definition

By looking for two quantified observables: Regularity and Complexity associated with the intuitive notion

Inspired by the recorded activity of cultured neural networks

We try to make sense out of the mess

On the Agenda

Cultured networks and their activity

Hints about self-regulation

The requirements from the new observables

Looking at the time-frequency plane

The best tilling

The sequence regularity

The structure factor and structural complexity

Results

Looking ahead

Our approach:Relative information in both time and frequency:

Tiling of the time-frequency plane

time

freq

uenc

y

V. What do we expect from a measure of Structural

complexity?[Hubberman and Hogg, Physica D 86, Gellmann “The Quark and the Jaguar”]

Comparison of the

Time-frequency domains

recorded shuffled

New clue

Local and global variations

Neuronal cell cultures

• Dissociated cell culture from the cortex of one-day-old rats.

• 10,000 neurons/mm2.

50µm

Multi Electrode Array

Non-Invasive Recording of the Activity(Capacitive Coupling between neurons and electrodes)

……. Polymer

30micro

20 milliseconds

one action potential of one neuron

Time

Formation of Bursting Events

Tracking a WOOZLE

Information-bearing templates

in the

Temporal ordering

of the

Recorded spontaneous activity

CONCISE HISTORICAL PERSPECTIVE

Neurons are binary elements

Localized information

storageDistributed information storage

RATE CODING vs. PULSE CODING

Currently: a Dynamic Networks picture

Guiding Questions

1. Is the spontaneous activity arbitrary or regulated ?

2. Can it provide clues about coding,

storage and retrieval of information ?

Statistical scaling properties of the SBE sequences

I(i)

Interval distribution Increment distribution

DifI(i)

Increment length (sec/τbin)

Pro

babl

ity

dens

ity

func

tion

(pd

f)

0 2 0

Lévy distribution

=2

1/1/ The sequence's plasticityThe sequence's plasticity

11// The sequence's regularityThe sequence's regularity

Comparison between networks of various sizes

1. Similar most probable interval ~10 sec

2. DifI(i) can be approximated

with zero-means symmetric Levy distributions

Small medium large50 20,000 1000,000

(higher density)

Experimental Model

100 sec 100 sec0.1 sec 0.1 sec

This feature can be simulated in modeled networks

if the neurons have two degrees of freedom

and the synapses are dynamical

Interfacing Real and Modeled Networks

Volman et al., Phys Rev E

1.Feeding the modeled network from regulating neurons

2.Testing the effect of synaptic strengths

conclusion

To show the same rate of activity as the similar networks

the large networks have to be

composed of coupled sub-networks

Hints about Self-Regulation

Controlled large variations vs. arbitrary large fluctuations

< [DifI(i)] >2

recorded

shuffled

model network

another hint : hierarchical temporal ordering

Bursts of SBEs , bursts of bursts of SBEs …

x10

Time cascade

1ms 100ms 5-10sec 500-100sec

Spike width SBE width Inter-SBEs Inter bursts of SBEs

510 210 [Hz]

Third hint :LONG-TIME CORRELATIONS

OVER a DAY !!!

THE OBSERVATIONS IMPLY THAT

Both the PULSE CODING

and the RATE CODING

do not provide the proper template

A new picture is needed

A DEDUCED CLUE

The recorded sequences

should be mapped

(via wavelet packet decomposition)

into time-frequency domains

time

freq

uen

cy

“energy”

time resolution

frequ

ency

resolution

Local and Global variations

Structural complexity

What have we seen so far?

Local features in segments of time series.

Temporal ordering and local rates.

Variation among segments.

Detour - Time-Frequency analysis

A. Wavelet Transform

0

2

0

d)(

C11

where

2ba,

ba,ba,

a

dadbtψ

aba,W

Ctf

a

b-tψtψdttψ

a

1f(t)ba,W

Detour - Time-Frequency analysis

A. Wavelet Transform

Time-Frequency Plane of the Wavelet Transform

Time bins

Time bins

Fre

quen

cy b

ands

Detour - Time-Frequency analysis

B. Wavelet Packets Decomposition

Coifman & Wicherhauser, 1993

How do we choose packets?

)m,n(II k,jm,n packets choose we , if

2

nn

)t(f

ψf(t)q

: npacket of energy the is qn

)qlog(q)qlog(qI

(n,m)

mmnnm,n

:n)informatio Shannon by (inspired

packetsfor functioncost The

Phase I:

Level 0 Level 1 Level 2

or ?Phase Ia: or ?Phase Ib: or ?Phase Ic: or ?Phase Id:

Phase II:

Level 0 Level 1

or ?Phase IIa:The best tiling:Thiele & Villemous, A.C.H.A., 1996

The Best Tiling Algorithm

Back to Structural complexity…

Physical Intuition - magnetization

tiles length of signal binbin NN

binbin

NarN

1;

Δt

Δωar :tile a of ratioaspect

1R1

Nlog

arlogR

n

bin2

2n

:tile a of resolution relative

binN

1nn

bin

RN

1RM :measure regularity

Regularity Measure

Structure factor

energy) zero-non with tiles only (counting

neighbors tilingnearest :

:factor structure

n,m

RRSFn,m mn

Structural complexity

jSF

)Nj1N

:word eachfor factor structure

( words into segmented is signal a

wN

1j

2

j SFSFN

1)SFvar(SC

:complexity structural

Our results:

Studied using artificial sequences with Levy distribution

The Regularity-Complexity Plane

Applying to neuronal data:Neuronal time series of SBEs Shuffled Neuronal time series

freq

uenc

y

timefr

eque

ncy

time

Zoom: shuffling of neuronal data

Finding a characteristic time scale

x10

Testing the Generality of Motives

Investigating cultured networks

made of neurons taken from

the frontal ganglion.

In vivo

In vitro

frontalganglion

Ex vivo

RATIONALE

This ganglion has a specificrole (feeding).

We will compare recordingsfrom the ganglion insidethe animal while feedingand while “thinking”, andwhen on the plate.

Looking for “function-follow-form” in action

The Statistical Scaling Parameters of

In-vitroEx-vivo

In-vivoIn-situ

“thinking”digesting

3 different neurons

regularity

0.8

0.75

0.7

0.65

0.6

0.55

0.5

0.45

0.4

1.61.8

1.4

2.12.0

6.0

In-vivo (f)

Ex-vivo

In-vivoCulture

Culture

gama alpha

com

ple

xit

y

Self-regulated complexity of neural activity

Hulata et al., PRLAyali et al., ?In vitro

In situ

In vivo (thinking)

Ex vivo

10sec

Spikes ,

4 days

20sec

Alternation between active and non-active phases .

5-6 days

Looking at Networks Development

100sec

Burst organization9-10 days

100sec

Hierarchical structureBurst of bursts,

14 days

100msec 10msec

Electrical activity Recorded from 14 days network

Probability density function of increments distribution

APs time series

T1 … Tn-1 , Tn , Tn+1 …

Inter-spikes Intervals

ISIn= Tn-Tn-1

Increments of ISI

(ISI)n = ISIn- ISIn-1

4 days 5-6 days 9 days

104 106102100msec

100 102 104msec 104 106102100

msec

log-log scale

linear scale

Inter burst intervals (IBI) distribution parameters

“young” networks “mature” networks

γ 2

5<δ<10

α1.6

“young” network Pdf parameters

γ 5

5<δ<10

α1.2

“mature” network Pdf parameters

Most probable interval

Decay slope

t=0.002

Structural complexity of burst time series

complexity regularity

“young” networks

“mature” networks

“mature” networks

“young” networks

Structural complexity of spike sequence during 4-6 days

0 9 183 6 12 15 0 9 183 6 12 15

time (hours) time (hours)

Complexity Regularity

306 12 18 240

time (hours)

306 12 18 240time (hours)

ComplexityRegularity

Developments of the Bursts Regularity-Complexity

γ 55<δ<10

α1.2

“mature” network Pdf parameters

γ 25<δ<10

α1.6

“young” network Pdf parameters

α 1/α;SC

γ 1/γ;RM

0=δ 20=δ

random periodic

ConclusionsWe have defined a new set of structural measures: Regularity Measure, Structure Factor, Structural Complexity.The measures fulfill both intuitive and quantitative requirements.The measures reveal new features of the neuro-informatic template of in-vitro neural networks.Future work: 1. comparison of in-vitro vs. in-vivo networks.2. study the effects of chemical substances,

coupling between networks, stimulations etc.

binN

1nn

bin

RN

1RM :measure regularity

Regularity Measure