An Embodied Developmental Robotic Model of Interactions between Numbers and Space

An Embodied Developmental Robotic Model of Interactions between Numbers and Space Marek Ruci nski (marek.rucinski@plymouth.ac.uk) Angelo Cangelosi (angelo.cangelosi@plymouth.ac.uk) Tony Belpaeme (tony.belpaeme@plymouth.ac.uk) Centre for Robotics and Neural Systems, University of Plymouth Plymouth, Devon, PL48AA, UK Abstract Changeux (1993). The system consisted of a 1-dimensional visual retina, a location and normalization cluster, a sum- mation coding layer and a place coding layer. The output from the place coding layer was used in a “same-different” comparison and “larger-smaller” comparison tasks. The sys- tem was designed to model perception and processing of non-symbolic stimuli (e.g. a cardinality of a set of items perceived visually). One of the most interesting findings was the demonstration of how the described system can autonomously “discover” the larger-smaller relation based solely on unsupervised experimentation with addition and subtraction. In this paper we describe an embodied developmental model of the interactions between the neural representations of num- bers and space in the humanoid robot iCub. We show how a simple developmental process that mimics real-world cultural biases leads to the emergence of certain properties of the num- ber and space representation system that enable the robot to reproduce well-known experimental phenomena. We demon- strate the validity of the proposed approach by showing that it leads to the reproduction of three psychological phenom- ena connected with number processing, namely size and dis- tance effects, the SNARC effect and the Posner-SNARC effect. Keywords: mathematical cognition; developmental cognitive robotics; computational modeling; size effect; distance effect; SNARC effect; Posner-SNARC effect; One of the first models of number representation based on recurrent artificial neural networks was proposed by Rodriguez, Wiles, and Elman (1999). Here, supervised learn- ing techniques were used to teach a simple recurrent neural network to perform a task, in which counting was required in order to succeed. In successful networks, neurons formed a special case of a discrete-time dynamical system in which numerosity was coded in the dynamical properties of trajec- tories realized by hidden layer units. This complex solution, radically different from traditionally used coding methods, has been obtained despite a small amount of inductive bias in the training process. Introduction Perceiving numbers and quantities is one of the most basic perceptual skills of humans and animals (Dehaene, 1997). Given the pure and abstract character of the number concept as perceived by humans, it is no surprise that many in cog- nitive science pursue a better understanding of how such a peculiar concept could have emerged, how it is represented and processed, and how it relates to other processes that take place in the brain. These efforts, which can be put together under the common label of mathematical cognition, consti- tute a branch of science that has been gaining more and more momentum during the past few decades (Dehaene & Bran- non, 2010). Computational modeling is an important tool used in the study of mathematical cognition to understand the principles of number processing in the brain. Based on observations from experimental psychological studies as well as hints ob- tained through various brain imaging techniques, computer models of number representation and processing are con- structed and evaluated on the basis of how well their proper- ties match those of the biological cognitive systems. Analysis of the computer models helps us to understand how biolog- ical systems work at the algorithmic level, which in turn is necessary to understand their neural implementations. In this paper we present an embodied developmental cogni- tive robotic model of interactions between number and space. In the following paragraphs we provide a short review of pre- vious computational models of numerosity representation and processing, focusing on those most relevant to the work pre- sented herein. An influential connectionist model of number represen- tation and processing has been described by Dehaene and Ahmad, Casey, and Bale (2002) presented a rather com- plex system aimed at modeling two manifestations of numer- ical abilities: subitizing and counting. Their system consisted of two networks, each specifically designed to perform one of these tasks, composed of several modules playing different roles and trained separately using various machine learning techniques. Implementation of the model delivered interest- ing results especially in the domain of counting (which, being a more complex task with a temporal structure, has been more rarely tackled in the literature than instant comprehension of numerosity), where counting error patterns similar to those observed in children were obtained. A relatively consistent path of increasingly complex mod- eling of different aspects of human mathematical cognition can be found in a series of papers by Verguts and collabora- tors (Verguts & Fias, 2004; Verguts, Fias, & Stevens, 2005; Gevers, Verguts, Reynvoet, Caessens, & Fias, 2006; Chen & Verguts, 2010). The first model focused on how sim- ple number coding methods believed to be employed in the brain (summation and place coding) can emerge as the re- sult of an unsupervised learning process, thus showing that