Cooperative Process

The winning neuron places the center of a topological neighborhood of cooperating neurons. Practically, a neuron that is firing tends to excite the neurons in its immediate neighborhood more than those farther away from this, which is intuitively satisfying. Such observation leads to make the topological neighborhood around the winning neuron i, decay smoothly along with lateral distance. To be exact let h_j,i indicate the topological neighborhood centered on winning neuron i, and encompassing a set of j or excited neurons. Assume d_i,j indicate the lateral distance between winning neuron i and excited neuron j. Then an assumption can be made that the topological neighborhood h_j,i is a uni-modal function of the lateral distance d_j,i, such that it satisfies two distinct needs as:

• The topological neighborhood h_j,i is symmetric to the maximum point defined by d_j,i = 0; in other words it gets its maximum value at the winning neuron i, for which the distance d_j,i is zero.
• The amplitude of the topological neighborhood h_j,i reduces monotonically along with the increasing lateral distance d_j,i, decaying to zero for d_j,i → ∞; it is necessary condition for convergence.

A typical choice of h_j,i such satisfies these needs is Gaussian function:

                                                       h_ij(x) = exp{-(d²_j,i)/2s²}..............................Eqn(15)

This is translational invariant

 The parameter σ is the "effective width" of topological neighborhood; this measures the degree to which excited neurons in the vicinity of the winning neuron participate in the learning method or process. This also makes the SOM algorithm converge faster than a rectangular topological neighborhood would.

For cooperation in between neighboring neurons to hold, this is essential that topological neighborhood be dependent on lateral distance between excited neuron j and winning neurons i in the output space quite than on some distance measure in the original output space. It is precisely what we have in Eq. 15. In case of one dimensional lattice, d_i,j is an integer equivalent to | i - j |. On the other hand, in case of a two dimensional lattice is explained by:

                                                            d²= || r_j  - r_i ||...................................Eqn(16)

Where the distance r_j shows the position of excited neuron j and r_i shows the discrete position of winning neuron i, both of this are measured in the discrete output space.

Other unique feature of the SOM algorithm is the size of the topological neighborhood shrinks along with time. It requirement is satisfied by making the width σ of the topological neighborhood function h_i,j decrease along with time. A popular alternative of the dependence of σ upon discrete time n is the exponential decay.

                                                           s(n) = s₀ exp(-(n/ t₁))..............................Eqn(17)

Whereas σ₀ is the value of σ at the initiation of the SOM algorithm and

And τ₁ is a time constant.