1 - Learning associations

The SOCR model uses two different learning equations for the strengthening and weakening of associations. Whenever two stimuli are contiguous, strengthening occurs. In such a case, the strengthening of the association from stimulus i to j after trial t, v_i, j^t is given by:

Δv_i, j^t = x_i^tα_iα_j(λ_j − v_i, j^t − 1)

where x_i^t denotes the presence (1) or absence (0) of stimulus i on trial t. As such, the SOCR model only learns about stimuli that are presented. The parameters α_i and α_j are the saliencies of stimuli i and j, respectively, and λ_j is the maximum association strength supported by j (the asymptote).

Whenever stimulus i is presented alone (i.e., stimulus j is absent), the weakening of that association is given by:

Δv_i, j^t = x_iα_i × −ω_jv_i, j^t − 1

where ω_j determines the weakening rate for stimulus j.¹

2 - Activating stimuli

SOCR posits competition by stimuli that are presented and/or associatively retrieved. Dropping the trial notation for the sake of simplicity, the degree to which stimulus i activates stimulus j, act_i, j, is given by:

act_i, j = x_iv_i, j + x_jρ_jα_j

where ρ_j (bound between 0 and +∞) determines how much of salience of stimulus j contributes to its unconditioned activation. These first-order activation values are the key quantities involved in the comparison processes.

3 - Generating responses and comparison processes

Stimulus i generates j-oriented responding at the time of retrieval as a function of its relative ability to activate stimulus j. This relative ability is expressed as a comparison process, given by:

r_i^j = act_i, j − Σ_{k ≠ i, j}^Kγ_k × o_i, k, j × r_i^k × r_k^j where r_i^j is the relative activation of stimulus j by stimulus i, K is the set of all experimental stimuli not including i or j, γ_k is a parameter determining the degree to which stimulus k, a comparison stimulus, contributes to the comparison process (bound between 0 and 1), and o_i, k, j is an operator switch that determines whether i and k associations with j engage in facilitation or competition. Finally, r_i^k is the relative activation of stimulus k by stimulus i, representing the ability of stimulus i to activate a comparison, and r_k^j is the relative activation of stimulus j by stimulus k, representing the ability of the comparison stimulus k to activate stimulus j.²

Most notably, the last two quantities (r_i^k and r_k^j) are also determined by their corresponding instantiations of Eq. 3. That is, they involve comparison processes themselves. The number of potential comparison processes is technically infinite (each comparison process can nest two extra comparison processes itself), so the user must determine the order of the model using an extra global parameter (order). For all n-th order models (with n > 0), the model will behave like the extended comparator hypothesis (Denniston et al., 2001), implementing n comparison processes each time the relative activations are calculated. With order = 0, SM2007 will behave like it was originally written and only consider one comparison process. Indeed, n-th order models are accomplished via recursion using the 0-th order model as the stopping condition. When such a condition is reached, the r_i^k and r_k^j terms in Eq. 3 become act_i, k and act_k, j, respectively.

4 - Switching between facilitation and competition

The operator switch in Eq. 3, o_i, k, j, changes as subjects learn to discriminate between the directly (via i) and indirectly activated (via k) representations of stimulus j. The change to this quantity depends on the value of v_i, j, as follows:

$$ \tag{Eq.4} \Delta o_{i,k,j} = \begin{cases} \tau_j\alpha_iv_{i,k}v_{k,j}(1-o_{i,k,j}) &\text{, if } v_{i,j} = 0\\ 1-o_{i,k,j} & \text{, otherwise} \end{cases} $$

where negative values of o indicate facilitation and positive values of o indicate competition. The default value for all operator switches at the outset of training is set as -1 by default. The parameter τ_j specifies the learning rate for the operator switches related to stimulus j.