# Neural Networks B, SS06 1

UA Dr. Robert Legenstein, WiMAus Prashant Joshi, M.S.

Institute for Theoretical Computer Science
Technische Universität Graz
A-8010 Graz, Austria
{legi, joshi}@igi.tugraz.at

 NACHNAME Vorname Matrikelnmr Team

# Exercise 1: Simulating a sigmoidal neuron with integrate-and-fire neurons

NOTE: You can download this exercise in pdf or postscript(ps) format here.

In this exercise you are going to simulate the output of a sigmoidal neuron using a feedforward circuit of intergrate-and-fire neurons. You are going to use the CSIM simulator for the purpose of this simulation.

# Theory

Figure 1 shows a neuron that receives R inputs. The individual inputs are each weighted by corresponding elements of the weight matrix W.

The neuron has a bias , which is summed with the weighted inputs to form the net input :

 (1)

This expression can be written in matrix form:

 (2)

where the matrix W for the single neuron case has only one row.

Now the output of this neuron can be written as

 (3)

For this exercise, let: , , and the transfer function of the neurons be the hyperbolic tangent sigmoid function that is defined as 2:

 (4)

which is called in MATLAB as the function.

In this problem, you have to simulate the output of the sigmoidal neuron with above mentioned properties using a feed-forward citcuit of integrate-and-fire neurons. An integrate-and-fire neuron is one of the simplest mathematical model of a spiking neuron whose membrane potential is given by:

 (5)

where is the membrane time constant, is the membrane resistance, is the current supplied by the synapses, is a non-specific background current and is a Gaussian random variable with a zero mean and a given variance noise.

At time is set to . If exceeds the threshold voltage it is reset to and held there for the length , which is the absolute refractory period of this neuron.

In principle, you can simulate the output of a sigmoidal neuron using a feedforward circuit of integrate-and-fire neurons. In this exercise we are going to test this. The setup that acheives this is shown in figure 2. For this, each of the integrate-and-fire neurons of the feedforward circuit receive the input , and the output of this circuit at any time point is defined as:

 (6)

where is the number of active neuron at time , and is the total number of neurons in the feedforward circuit. This is an example of population coding, where the value of a variable at any point of time is encoded by the ratio of number of active neurons in the population to the total size of population.

# Problem

Let the inputs to the sigmoidal neuron be a 10 dimensional vector, such that . Let the feedforward circuit of integrate-and-fire neurons have neurons. Simulate the functioning of this neuron using the setup shown in figure 2 for the case when the weight vector from the inputs to the sigmoidal neuron has the values:
1. ( points) = ones(1,ipDim)/ipDim; % (ipDim = 10)
(ipDim is the size of a single input pattern).
2. ( bonus points) = randperm(ipDim); %(ipDim = 10)
= w/sum(w);

For this you will have to simulate the circuit once per input pattern followed by a reset.

NOTE: You are also required to submit the source code that you used to do this exercise.

Neural Networks B, SS06 1

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#### Footnotes

... SS061
Class Website: http://www.igi.tugraz.at/lehre/NNB/SS06/
... as2
We only consider positive activation of this function as all elements of .

Joshi Prashant 2006-03-28