Www.gfai.de/~heinz How Network Topology Defines its Behavior - Serial Code Detection with Spiking Networks Dr. Gerd Heinz Gesellschaft zur Förderung angewandter

ww

w.gfai.de/~

heinzw

ww

.gfai.de/~heinz

How Network Topology Defines its Behavior How Network Topology Defines its Behavior --Serial Code Detection with Spiking Serial Code Detection with Spiking NetworksNetworks

Dr. Gerd Heinz Dr. Gerd Heinz

Gesellschaft zur Förderung Gesellschaft zur Förderung angewandter Informatik e.Vangewandter Informatik e.V

Berlin-AdlershofBerlin-Adlershof

Workshop „Autonomous Workshop „Autonomous Systems”Systems”

Herwig Unger & Wolfgang HalangHerwig Unger & Wolfgang Halang

Hotel Sabina Playa, Cala Millor Hotel Sabina Playa, Cala Millor

Mallorca, 13-17 Oct. 2013Mallorca, 13-17 Oct. 2013 Sensor- und Motor- Sensor- und Motor- Homunculus. Homunculus.

Natural History Museum, Natural History Museum, London London

ww

w.gfai.de/~

heinzw

ww

.gfai.de/~heinz

Contents

Abstract Convolution A Small Interference Network Construction of Transfer Functions Applying a Convolution Spike Output Frequency Analysis Unipolar or Bipolar Signal Levels? Interpreting Bursts Examples

[email protected] www.gfai.de/~heinz

Abstract Compared with technical sensors, sound and code analysis of

nerve system is fascinating We differ between the whisper of the wind or the branding of

waves, we know the songs of birds, we hear dangerous noises of a defect car engine, we feel, if an airplane starts

And we speak and understand languages: Do we have a chance, to interprete the function of a nerve net on the level of net structure?

We try to analyze a simplest delaying network in nerve-like structure

Our net consists of delays T and weights W Basing on Interference Network (IN) abstraction we transform

the net into a transfer function H of a linear time-invariant system (LTI-system)

We use convolution between input time-function and transfer function to find the "behaviour" of the LTI-system

* The work bases on the book "Neuronale Interferenzen", Kap.8b, S.181, download: www.gfai.de/~heinz/publications/NI/index.htm


Convolution

"Faltung" (terminus created by Felix Bernstein, 1922):

Discrete form (Cauchy product):

Example: FIR-filteras direct implementationof convolution, form: Y = X * S

dhtxthtxtyt

)()()(*)()(0

kn

n

kkn xhy

0


A Small Interference Network

N N'

x(t) y(t)

1

2

n

. .

.

1w2w

nw

. . . +

N N'

x(t) y(t). .

.

Form: Our Abstraction:

Delay vector:

Weight vector:

Transfer function:

],...,,[ 21 nT

],...,,[ 21 nwwwW

n

iii txw

nty )(

1)(


Construction of Transfer Function H

(Transfer function of LTI-system)Discrete transfer function H seen as discrete time function with

sample distance ts = 1/fs and with growing index i :

i = [… 2, 3, 4, 5, 6, 7, 8, 9, …]

H = [… wi-1, wi, wi+1, wi+2, wi+3, wi+4, wi+5, wi+6, …]

Length of H is greater the delay difference: length(H) ≥ max(T) – min(T)

Construction of the transfer function of the net by addition of weights:H(j) = H(j) + W(i) mit j = T(i) :H(T(i)) = H(T(i)) + W(i)

fs = 1/ts


Get Transfer Function with Scilab

function [H] = trans(T,W,fs); if length(T) == length(W) then T = T * fs; // apply sample rate of H T = round(T); // T becomes index: integer H = 1:max(T); H = H * 0; // create an empty H for i = 1:length(T), // for all T(i), W(i) j = T(i), // delay becomes the H-index j H(j) = H(j) + W(i), // add the weight to H end // for else // if printf('\n\nerror: T and W have different size\n'); end // if endfunction;

H is the transfer function of a LTI-system!


Applying a Convolution

What is the system answer Y for different input functions X ?It is simple the convolution with H , the multiplication of time series

y(t) = h(t) * x(t)

Using vectorsY = X * H

Scilab formY = convol(H,X)

Fourier Analysis of H F = abs(fft(H))

HX Y


Barker Codes and Spikes

Hebbian rule in neuro-science shows, that a neuron needs high synchronous emissions to learn

We need spikes at the output of the neuron Barker codes maximize spike-like output of long

sequences in RADAR technology:

Example:H = [1, 1, 1, -1, 1] (Barker code no. 5)X = rev(H)

Y = convol(X,H) = [1, 0, 1, 0, 5, 0, 1, 0, 1]

But neurons don't have negative signal values!What can we do?


Spectral Analysis of Transfer Function H

FFT of any unipolar transfer function shows the maximum for frequency f = 0 Hz (DC)

It is not possible to learn with unipolar H ; codes are AC:

Unipolar{0…1}

Bipolar{-1…1}

n

njj enHeF )()(

Highest level at 0 Hz


Unipolar or Bipolar Signal Levels?

Unipolar signals, unipolar synapses: {-1…0…1}



Bipolar signals, bipolar synapses: {0…1}



Unipolar signals, bipolar synapses (neuron) {0…1} {-1…1}



unipolar signals and bipolar synapses (neuron) X, Y: uni {0…1}

H: bi {-1…1}

Big surprize: Using unipolar signals X, Y and bipolar H, the system is not

significant worse compared to the best case uni/uni

Test it: Use relating Scilab sources under

www.gfai.de/~heinz/publications/papers/2013_autosys.pdfwww.gfai.de/~heinz/techdocs/index.htm#conv

Conclusion Nerve systems do not need bipolar signals to detect code and

sound, if the synapses are bipolar (inhibiting or exciting)!


Interpreting Bursts

Noisy groups of pulses are known at different locations in nerve system

Is it possible, to find the net structure behind them?


The Inversive Procedure

We interprete a burst as transfer function H (seen as pulse response) and reproduce the delays T and weights W of the network behind:function [T,W] = net(H,fs); // returns T and W j=1; // W-index j for i=1:length(H) // H-index i if H(i) == 0 then ; // do nothing else // write the value to W, the index to T W(j) = H(i); // value to W T(j) = i; // index to T j = j+1; // increment j end; // endif end; // endfor T = T ./ fs; // multiply with sample duration T = T - min(T); // scale to min: reduced T-vectorendfunction;


Example H = f(T,W)

Delays T, weights W, transfer function H, reducing vectors: index r

Delays:Weights:

Reduced T, W:

Transfer function:

]8,3,5[],,[ 321 T

]1,5.0,1[],,[ 321 wwwW

]5,2,0[],,[ 321 RRRRT

]1,1,5[.],,[ 321 RRRR wwwW

),0,0,,0,( 312 wwwH

1,0,0,1,0,5.0H


Example

Key X and keyhole Hunipolar

max(FFT) at 0 Hz(uni/uni)


Conclusion

To characterize time- and frequency domain, we transform delays and weights of a simplest interference network into a LTI transfer response

A procedure [H] = trans(T,W,fs) calculates the (time-discrete) transfer function H (pulse response) of the net from delay vector T (delay mask) and weight vector W

The FFT shows learning problems for unipolar signals and unipolar H because of highest DC-value

A mixture between unipolar signals and bipolar transfer function (weights) acts as good alternative (nerve nets)

Interpreting bursts as transfer functions (pulse responses), we design an inverse procedure [T,W] = net(H,fs) that reconstructs the net structure [T,W] from transfer function H

Find Scilab sources and the paper on the webwww.gfai.de/~heinz/publications/papers/2013_autosys.pdfwww.gfai.de/~heinz/techdocs/index.htm#conv


Relevance for ANN

The transfer function or pulse response H is responsible for all sequential properties of a network: for code and sound generation or detection

The lecture shows, that smallest delays and delay differences change the pulse response H of the network

Remembering the "Neural Networks" (NN, ANN) approach with layers clocked by clock cycles we find, that the NN-approach destroyes the sequential structure of each network complete

In no case ANN or NN are candidates to understand the function of nerve like structures

Thinking about nerves we need interferential approches that does not destroy the delay structure of the net.


Vielen Dank für die Vielen Dank für die Aufmerksamkeit!Aufmerksamkeit!

Dr. G. Heinz, GFaIDr. G. Heinz, GFaI

Volmerstr.3 Volmerstr.3

12489 Berlin12489 Berlin

Tel. +49 (30) 814563-490Tel. +49 (30) 814563-490www.gfai.de/~heinz

[email protected]

Erfolgreiche Google-Suchterme: Erfolgreiche Google-Suchterme: "Interferenznetze", "Mathematik des "Interferenznetze", "Mathematik des Nervensystems", "Heinz", Nervensystems", "Heinz", "Akustische Kamera""Akustische Kamera"

Und der Herr sprach: "So führte ich Und der Herr sprach: "So führte ich euch auf den Weg der Erkenntnis. euch auf den Weg der Erkenntnis. Gehet nun, und traget die Botschaft in Gehet nun, und traget die Botschaft in die Welt hinaus!"die Welt hinaus!"

Documents

Www.gfai.de/~heinz How Network Topology Defines its Behavior - Serial Code Detection with Spiking Networks Dr. Gerd Heinz Gesellschaft zur Förderung angewandter