Replicator dynamics of speech perception and categorization

A central question in cognitive modeling is whether different levels of representation influence one another interactively, through bidirectional feedback or resonance (e.g., Grossberg, 1976, 2003; McClelland & Elman, 1986; Elman & McClelland, 1988; Massaro & Cohen, 1991; Magnuson et al., 2003, Rapp & Goldrick, 2000, 2004). In this paper, we demonstrate that interactivity is not needed to account for the detailed, time-dependent pattern of speech perception and categorization found in recent eye-tracking experiments (McMurray & Spivey 2000). Given minimal assumptions, a non-interactive mathematical model based on the replicator equation of evolutionary dynamics (Nowak 2006) correctly predicts both traditional behavioral findings of speech identification, and that categorization over time displays an evolving sigmoid pattern. This model can be instantiated as a novel type of connectionist network in which the activity of a unit is updated by multiplying its current value and the sum of the incoming excitation. Unlike many proposed models, the replicator network does not depend on the unverified claim that acoustic or auditory representations are affected by top-down feedback from phoneme, lexical, or other levels. This paper contributes to the formal understanding of the role and necessity of interactivity, the influence of both phonemic and acoustic information, and is (as far as we are aware) the first to apply the parsimonious formal methods of evolutionary dynamics to the domain of on-line speech perception. McMurray & Spivey (2000) presented participants with a 9-step /pa/-to-/ba/ continuum in which Voice Onset Time (VOT) ranged from -50ms to +60ms in equal-sized steps of approximately 12ms. Participants heard the stimuli while viewing a computer display with the two response possibilities (/pa/ and /ba/) in predictable locations. Mouse-choice categorizations and eye-gaze movements and locations were recorded from stimulus offset. The distribution of eye gazes at the latest time period is essentially indistinguishable from the distribution of mouse choices; both show the sigmoid-shape curve that is characteristic of speech categorization experiments, with a sharp category boundary at approx. +10ms. The eye-gaze data at earlier time periods shows a gradual evolution from a relatively ‘flat’ profile, which departs only slightly from chance looking, through a series of increasingly ‘sharp’ sigmoids that culminates in the final categorization distribution (see Fig. 1, reproduced from McMurray & Spivey, 2000: Figure 7). Extensive curve-fitting by McMurray and Spivey suggests that, at every measured point in processing, the eye-gaze responses have a sigmoid shape with pivot point at the category boundary. They call the temporal evolution from flat to sharp gaze distributions, observed for the first time in their experiment, the evolving sigmoid. Our lab is in the process of collecting relevant patient data, as well as extending McMurray & Spivey’s important finding to a different (vowel quality) continuum. McMurray & Spivey (2000) propose an interactive connectionist model, the Hebbian Normalized Recurrence Network, that succeeds in capturing the evolving sigmoid. Similar in its mechanics to an iteratively (re)normalized version of McClelland & Elman’s TRACE or Grossberg’s ART model, this network has two qualitative properties that McMurray & Spivey identify as crucial to its success. First, the network is sensitive to the statistical structure of the two categories /p/ and /b/ in virtue of competitive Hebbian learning; we approximate the statistical structure as two widely-separated normal distributions that give the probability densities of VOT values in each category. Second, the network displays competitive processing that plays out over time; the probability of responding with one category or the other changes during the competition in a way that mimics the experimentally observed sigmoid evolution. McMurray & Spivey’s modeling results appear to support interactivity (feedback from the phoneme level of representation to the acoustic/auditory level) in speech perception and categorization. However, the replicator equation (Nowak, 2004) is a straightforward embodiment of time-dependent competitive processing that does not require interactivity/feedback between levels of representation. The equation is x& i = xi ⋅ (fi – φ), where φ = Σj (xj ⋅ fj). In biological applications, xi denotes the proportion of the Laboratory Phonology 11 163