**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 |

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

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.

- ( points) = ones(1,ipDim)/ipDim; % (ipDim = 10)

(ipDim is the size of a single input pattern). - ( 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.

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- ... SS06
^{1} - Class Website: http://www.igi.tugraz.at/lehre/NNB/SS06/
- ... as
^{2} - We only consider positive activation of this function as all elements of .

Joshi Prashant 2006-03-28