3.2 The RF-LISSOM Architecture

**Figure 3.1:** **Architecture of the RF-LISSOM network.**
A tiny RF-LISSOM network and retina are illustrated, along with connections to a single neuron (shown as a large circle). Other neurons have similar connections to other areas of the retina and to the neurons surrounding them. The afferent connections form a local anatomical receptive field on the simulated retina. Neighboring neurons have different but highly overlapping RFs. Each neuron computes an initial response as a dot product of its receptive field and its weight vector. The responses then repeatedly propagate within the cortex through the lateral connections (only a very small connection radius is shown) and evolve into an activity ``bubble''. After the activity stabilizes, weights of the active neurons are adapted.
$\begin{figure} \begin{minipage} {\textwidth} \centering \hputpictype{\pscolor} rf-lissom-architecture (0.8\textwidth) \end{minipage} \end{figure}$

The RF-LISSOM model is based on a simulated network of neurons with afferent connections from the external world and recurrent lateral connections between neurons. Connections adapt based on correlated activity between neurons. The result is a self-organized structure where afferent connection weights form a map of the input space, and lateral connections store long-term correlations in neuronal activity.

In RF-LISSOM, the cortical architecture has been simplified and reduced to the minimum necessary configuration to account for the observed phenomena. Because the focus is on the two-dimensional organization of the cortex, each ``neuron'' in the model cortex corresponds to a vertical column of cells through the six layers of the human cortex. This columnar organization helps make the problem of simulating such a large number of neurons tractable, and is viable because the cells in a column generally fire in response to the same inputs (chapter 2). Thus RF-LISSOM models biological mechanisms at an aggregate level, so it is important to keep in mind that RF-LISSOM ``neurons'' are not strictly identifiable with single cells in the human cortex.

The cortical network is modeled with a sheet of interconnected neurons and the retina with a sheet of retinal ganglion cells (figure 3.1). Neurons receive afferent connections from broad overlapping patches on the retina. The $N \times N$ network is projected on to the retina of $R \times R$ ganglion cells, and each neuron is connected to ganglion cells in a circular area of radius r around the projections. Thus, neurons at a particular cortical location receive afferents from the corresponding location on the retina. Depending on its location, the number of afferents to a neuron varies from roughly $r \times r$ (at the corners) to $2r \times 2r$ (at the center).

In addition, each neuron has reciprocal excitatory and inhibitory lateral connections with itself and other neurons. Lateral excitatory connections are short-range, connecting each neuron and its close neighbors. Lateral inhibitory connections run for comparatively long distances, but also include connections from the neuron and its immediate neighbors to itself. Thus the ``lateral'' connections in the model are not exclusively from neurons located laterally, since they include self-recurrent connections.

The input to the model consists of 2-D patterns of activity representing retinal ganglion cell activations. Each ganglion cell is modeled only by its activation levels, not by its receptive field, so the input pattern is equivalent to an image after it has been processed by the retina. In addition, the transformation of retinal activation patterns by the LGN has been bypassed for simplicity, since LGN neurons do not change the shape of the receptive fields of the retina (chapter 2). Thus the ``retina'' of the model could equivalently be considered to represent a pattern of activity across neurons in the retinotopic map of the LGN.

The RF-LISSOM network will self-organize to represent the most common features present in the input images it has seen (Sirosh, 1995). Since tilt aftereffects appear to arise in the areas processing oriented inputs (chapter 2), simple oriented inputs (two-dimensional Gaussians) were used in the experiments presented in this thesis, as described in more detail in chapter 4.

The weights are initially set to random values or a smooth distribution (such as a Gaussian profile), and are organized through an unsupervised learning process. At each training step, neurons start out with zero activity. The initial response $\eta_{ij}$ of neuron (i,j) is calculated as a weighted sum of the retinal activations: $\begin{displaymath} \eta_{ij} = \sigma \left( \sum_{a,b} \xi_{ab} \mu_{ij,ab} \right),\end{displaymath}$
where $\xi_{ab}$ is the activation of retinal ganglion (a,b) within the anatomical RF of the neuron, $\mu_{ij,ab}$ is the corresponding afferent weight, and $\sigma$ is a piecewise linear approximation of the sigmoid activation function (figure 3.2): $\begin{displaymath} \sigma(x) = \left\{ \begin{array} {ll} 0 & \mbox{$x \leq \d... ... x < \beta$}\\ 1 & \mbox{$x \geq \beta$} \end{array} \right .\end{displaymath}$

**Figure 3.2:** **The RF-LISSOM neuron activation function.**
The neuron requires an input as large as the threshold $\delta$ before responding, and saturates at the ceiling $\beta$ . The output activation values are limited to [0,1] . The activation function is an easy-to-compute approximation of the sigmoid function.
$\begin{figure} \centering \putpictype{\psline} sigmoid (2.25in) \end{figure}$

Lateral interaction in the cortex sharpens neuronal response by repeated exchange of activation (Kohonen, 1989; Mountcastle, 1968). In RF-LISSOM, the initial response evolves over a very short time scale through lateral interaction. At each time step, the neuron combines the above afferent activation $\sum \xi \mu$ with lateral excitation and inhibition: $\begin{displaymath} \eta_{ij}(t) = \sigma \left( \sum \xi \mu + \gamma_e \sum_{... ...(t-1) - \gamma_i \sum_{k,l} I_{ij,kl} \eta_{kl}(t-1) \right) ,\end{displaymath}$
where E_ij,kl is the excitatory lateral connection weight on the connection from neuron (k,l) to neuron (i,j) , I_ij,kl is the inhibitory connection weight, and $\eta_{kl}(t-1)$ is the activity of neuron (k,l) during the previous time step. All connection weights are positive. The scaling factors $\gamma_e$ and $\gamma_i$ determine the relative strengths of excitatory and inhibitory lateral interactions. While the cortical response is settling, the retinal activity remains constant.

The activity pattern starts out diffuse and spread over a substantial part of the map, but within a few iterations of equation 3.3, converges into a small number of stable focused patches of activity, or activity bubbles. This sharpens the contrast between areas of high and low activity, helping it become focused around the maximally responding area.

After the activity has settled, the connection weights of each neuron are modified. Both afferent and lateral weights adapt according to the same mechanism: the Hebb rule, normalized so that the sum of the weights is constant: $\begin{displaymath} w_{ij,mn}(t+\delta t)= \frac{w_{ij,mn}(t) + \alpha \eta_{ij}... ...um_{mn} \left[ w_{ij,mn}(t) + \alpha \eta_{ij} X_{mn} \right]},\end{displaymath}$
where $\eta_{ij}$ stands for the activity of neuron (i,j) in the final activity bubble, w_ij,mn is the afferent or lateral connection weight ( $\mu$ , E or I ), $\alpha$ is the learning rate for each type of connection ( $\alpha_A$ for afferent weights, $\alpha_E$ for excitatory, and $\alpha_I$ for inhibitory) and X_mn is the presynaptic activity ( $\xi$ for afferent, $\eta$ for lateral). Afferent inputs, lateral excitatory inputs, and lateral inhibitory inputs are normalized separately.

Following the Hebbian principle, the larger the product of the pre- and post-synaptic activity $\eta_{ij} X_{mn}$ , the larger the weight change. Therefore, when the pre- and post-synaptic neurons fire together frequently, the connection becomes stronger. Both excitatory and inhibitory connections strengthen by correlated activity; normalization then redistributes the changes so that the sum of each weight type for each neuron remains constant. In effect, such a rule distributes the weights of each neuron $\eta_{ij}$ in the proportion of its activity correlations with other neurons $\eta_{kl}$ , k,l=1..N .

At long distances, very few neurons have correlated activity and therefore most long-range connections eventually become weak. The weak connections can be eliminated periodically by the researcher; through the weight normalization, this will concentrate the inhibition in a closer neighborhood of each neuron. The radius of the lateral excitatory interactions starts out large, but as self-organization progresses, it is decreased (according to a schedule set by the researcher) until it covers only the nearest neighbors. Such a decrease is necessary for global topographic order to develop and for the receptive fields to become well-tuned at the same time (for theoretical motivation for this process, see Kohonen 1982, 1989, 1993; Obermayer et al. 1992; Sirosh and Miikkulainen 1997; for neurobiological evidence, see Dalva and Katz 1994; Hata et al. 1993.) Together the pruning of lateral connections and decreasing excitation range produce activity bubbles that are gradually more focused and local. As a result, weights change in smaller neighborhoods, and receptive fields become better tuned to local areas of the retina.