The activation frequency self-organizing map (AFSOM) Antonio Neme, Pedro Miramontes Universidad Aut´ onoma de la Ciudad de M´ exico San Lorenzo 290 Col. Del Valle Mexico, D.F. Mexico. neme@nolineal.org.mx Universidad Nacional Aut´ onoma de M´ exico keywords: self-organization, Hebbian learning, non-radial influence Abstract— In the self-organizing map (SOM), the best matching units (BMUs) affect neurons as a function of distance and the learning parameter. Here we study the effects in SOM when a new parameter in the learning rule, the activation frequency, is included. This parame- ter is based on the relative frequency by which each neu- ron is included in each BMU’s neighborhood, so there is an individual memory (synapse strength) of the activation received from each neuron. The parameter leads to non- radial influence areas for BMUs that modifies the map for- mation dynamics, including the fact that the weight vector for BMU may not be the closest one to the input stimulus after weight adaptation. Two error measures are lower for the maps trained with this model than those obtained with SOM, as shown in experiments with six data sets. 1 Introduction The self-organizing map (SOM) is presented as a model for the self-organization of neural connections, which is translated in the ability of the algorithm to produce organization from disorder [5]. One of the main properties of the SOM is its ability to preserve in the output map the topographical relations present in input data [16], This property is achieved through a transformation of an incoming signal pattern of arbitrary dimension into a low-dimensional discrete map and to adaptively transform data in a topologically ordered fashion [16]. Each input data is mapped to a BMU, which affects other neurons accordingly to the learning equation: w n (t + 1) = w n (t)+ α n (t)h n (g,t)(x i - w n (t)) (1) Where α(t) is the learning rate at time t and h n (g,t) is the neighborhood function from BMU neuron g to neuron n at time t. In general, the neighborhood function decreases monotonically as a function of the distance from neuron g to neuron n. This decreasing property has been stated to be a necessary condition for convergence [7, 6]. The SOM tries to preserve relationships of input data by starting with a large neighborhood and reducing it during the course of training [16]. Neighborhood and learning parameters are reduced by an annealing scheme although the form they take is not critical [16]. As pointed out in [29], SOM follows the idea of using a deformable lattice to transform data similarities into spatial relationships. The lattice is deformed by applying learning equation (1) to the neurons in the network. Here, we pro- pose an additional parameter for equation (1) that quanti- fies the influence a BMU n has over the neurons in the net- work as a function of the relative frequency with which n includes them in its neighborhood. It also measures the in- fluence each input vector m has on them as a function of the number of times the BMU for m affects the neurons. This frequency activation parameter allows non-radial neighbor- hood and, as reported in the results, forms better maps, in terms of two error measures. Although several modifications have been proposed to the SOM learning rule, they do not reflect, at least to our knowledge, the frequency of activation from other neurons. Some works incorporate non-radial influence from BMUs to neighbors, as, for example, in [19] it is proposed the re- cursive Fisherman’s rule and some hybrid rules that reflect an attenuation of the adaptation as the distance from the BMU to the affected neuron increases. In [12] the activ- ity patterns are non-radial and determined by a mechanism based on a cooperative information control. One of the first works that incorporated the concept of memory for each neuron was proposed in [4], in which an activation memory is defined, in order to identify the new active neuron, and a modification in the BMU selec- tion mecanism is presented. Also, a SOM-related model has been studied in the light of reaction-diffusion mecha- nisms where the BMU perturbes the excitable media and generates a symmetrical traveling wave [26]. Several mod- els of the visual cortex have been proposed [23, 21, 1] in which a dynamic Hebbian-like behavior is considered, but there is a radial influence between neurons. The activity patterns in the cortex are irregular and non- radial [13, 3], that is, neurons in different regions become active for a given stimulus, whereas SOM shows regular activity, since the neighborhood function defines a sym- metrical influence area, as well as the learning function, although there are some variations that present a different behavior. There is biological evidence that connectivity in the brain cortex is not regular and could be approximated by a small-world topology [31, 30], which means that when a given neuron or group of networks become active, they