The role of sparsely distributed representations in familiarity recognition of verbal and olfactory materials

Abstract

We present the generalized signal detection theory (GSDT), where familiarity is described by a sparse binomial distribution of binary node activity rather than by normal distribution of familiarity. Items are presented in a distributed representation, where each node receives either noise only, or signal and noise. An old response (i.e., a “yes” response) is made if at least one node receives signal plus noise that is larger than the activation threshold, and item variability is determined by the distribution of activated nodes as the threshold is varied. A distinct representation leads to better performance and a lower ratio of new to old item variability, than a more distributed and less distinct representations. Here we apply the GSDT to empirical data on verbal and olfactory memory and suggest that verbal memory relies on a distinct neural item representation, whereas olfactory memory has a fuzzy neural representation leading to poorer memory and inducing a larger ratio of new to old item variability.

Appendix

Mathematical description of UVSDT

Unequal-variance signal detection theory (UVSDT) is implemented as a cumulative normal distribution (Norm), where the probability of responding old (p_norm) is:

$$p_{\text{norm}} = \, 1{-}{\text{Norm}}\left( {t,S,V} \right)$$

where t is the threshold, S is the mean value of the distribution that the subject responds to, and V is the variability of this distribution. The subscript v is used for verbal responses and the subscript l for olfactory responses; the subscript o is for old items and n for new items. S_n = 0, S_o > 0, V_n = 1, V₀ > 1, and V_o,v > V_o,l. The model can be fitted by adjusting S, t, and V.

Parameters were selected by fitting the model to empirical data using the fminsearch function in MATLAB. The following parameters are used in the paper; S = 4.7, a_v = 1.4 (verbal data) and a_l = 3.4 (olfactory data), N = 36. The t value was set so that the false-alarm rate was symmetrical to the hit rate (i.e., 1 − p(H) = p(FA)). This is done by numerically solving the model with this constraint, where the value of t depends nonlinearly on other parameters in the model, i.e., particularly on the values of a and S.

Mathematical description of GSDT

Generalized signal detection theory is implemented by a cumulative binomial distribution (B):

$$p_{b} = \, 1 \, {-} \, B \, \left( {t_{b} ,N,p_{norm} } \right)$$

where p_b is the probability of responding old, t_b = 0 is the number of nodes required for a new response, and N is the total number of nodes. p_norm is the probability of a single node being active (as described above in SDT), where S_n = 0, S_o > 0, and V_o = V_n = 1. Thus, for new items p_{norm, n} = 1 − Norm (t, 0, 1), and for old items, it is the weighted value of nodes receiving and nodes not receiving a signal: p_{norm, o} = 1 − (a × Norm (t, S_o/a, 1) + (a − N) × Norm (t, 0, 1))/N. Thus, the model can be fitted by adjusting S, t, a, and N.

To simplify model fitting, N can take on continuous values (e.g., 36.3), where we weighted the binomial distributions with how close they were to the nearest integer value:

$$p_{b} = \, \left( {1 \, {-} \, \left( {N{-}N_{i} } \right)} \right) \times \left( {1 \, {-} \, B \, \left( {t_{b} ,N_{i} ,p_{n} } \right)} \right) \, + \, \left( {N{-}N_{i} } \right) \times \left( {1 \, {-} \, B \, \left( {t_{b} ,N_{i} + \, 1,p_{n} } \right)} \right)$$

where N_i is the highest integer value that is lower than N.