Tuesday, May 20, 2008

Artificial Neural Model

Artificial Neuron Model
As it is mentioned in the previous section, the transmission of a signal from one neuron to
another through synapses is a complex chemical process in which specific transmitter substances are released from the sending side of the junction. The effect is to raise or lower the electrical potential inside the body of the receiving cell. If this graded potential reaches a threshold, the neuron fires. It is this characteristic that the artificial neuron model proposed by McCulloch and Pitts, attempt to reproduce. The neuron model shown in Figure 1.6 is the one that widely used in artificial neural networks with some minor modifications on it.
Figure 6. Artificial Neuron
The artificial neuron given in this figure has N input, denoted as u1, u2, ...uN. Each lineconnecting these inputs to the neuron is assigned a weight, which are denoted as w1, w2, .., wN respectively. Weights in the artificial model correspond to the synaptic connections in biological neurons. The threshold in artificial neuron is usually represented by θ and the activation corresponding to the graded potential is given by the formula: The inputs and the weights are real values. A negative value for a weight indicates an inhibitory connection while a positive value indicates an excitatory one. Although in biological neurons, θ has a negative value, it may be assigned a positive value in artificial neuron models. If θ is positive, it is usually referred as bias. For its mathematical convenience we will use (+) sign in the activation formula. Sometimes, the threshold is combined for simplicity into the summation part by assuming an imaginary input u0 =+1 and a connection weight w0 = θ. Hence the activation formula becomes:

The output value of the neuron is a function of its activation in an analogy to the firing frequency of the biological neurons:
x = f (a) Furthermore the vector notation
is useful for expressing the activation for a neuron. Here, the jth element of the input vector u is uj and the jth element of the weight vector of w is wj. Both of these vectors are of size N. Notice that, wTu is the inner product of the vectors w and u, resulting in a scalar value. The inner product is an operation defined on equal sized vectors. In the case these vectors have unit length, the inner product is a measure of similarity of these vectors.
Originally the neuron output function f(a) in McCulloch Pitts model proposed as threshold function, however linear, ramp and sigmoid and functions (Figure 6.) are also widely used output functions:

Figure 7. Some neuron output functions
Though its simple structure, McCulloch-Pitts neuron is a powerful computational device. McCulloch and Pitts proved that a synchronous assembly of such neurons is capable in principle to perform any computation that an ordinary digital computer can, though not necessarily so rapidly or conveniently .

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