Neural Cognitive Modelling: A Biologically Constrained Spiking Neuron Model of the Tower of Hanoi Task

Neural Cognitive Modelling: A Biologically Constrained Spiking Neuron Model of the Tower of Hanoi Task Terrence C. Stewart (tcstewar@uwaterloo.ca) Chris Eliasmith (celiasmith@uwaterloo.ca) Centre for Theoretical Neuroscience, University of Waterloo 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada Abstract In previous work, we used this approach to create models of list memory (Choo & Eliasmith, 2010), rule induction (Rasmussen & Eliasmith, 2010), the Wason card task (Eliasmith, 2005), and action selection (Stewart, Choo, & Eliasmith, 2010a). We have also argued that our action selection model (based on the basal ganglia) is sufficient to implement the basic functionality of a production system (Stewart & Eliasmith 2010). This paper expands on this work to present the first spiking neural model implementing rule following, planning, and goal recall. The model conforms to the known anatomy, connectivity, and neural properties of the basal ganglia, thalamus, and cortex. The components of the model are general-purpose, in that they could be used to perform different tasks without changing any neural connections within the cortex. To define the task to be performed, we only need to set the synaptic connection weights between the cortex and basal ganglia and between the thalamus and the cortex. This makes our work both a neural explanation of a particular high-level cognitive task, and a general set of principles for creating neural models for other such tasks. We present a computational model capable of solving arbitrary Tower of Hanoi problems. All elements except visual input and motor output are implemented using 150,000 LIF spiking neurons. Properties of these neurons (firing rate, post-synaptic time constant, etc.) are set based on the neurons in corresponding areas of the brain, and connectivity is similarly constrained. Cortical components are all general- purpose modules (for storing state information and for storing and retrieving short-term memories of previous state information), and could be used for other tasks. The only task-specific components are particular synaptic connection weights from cortex to basal ganglia and from thalamus to cortex, which implement 19 context-specific rules. The model has a single free parameter (the synaptic connection weights of the input to short-term memory), and produces timing behaviour similar to that of human participants. Keywords: Tower of Hanoi; neural engineering; cognitive architectures; computational neuroscience Neural Cognitive Models To explain human behaviour, cognitive scientists must identify both what the brain does and how it does it. This involves finding the algorithms underlying cognitive performance as well as determining how these algorithms are implemented within the brain through the interaction of neurons, neurotransmitters, and other physical components. Our ongoing research is in the construction of large-scale neural models capable of exhibiting complex cognitive behaviour such as planning, rule-following, and symbolic reasoning. The intent is to bridge the gap between cognitive theory and neuroscience, allowing the fields to interact in both directions. With such a bridge, high-level cognitive theory would produce detailed low-level predictions as to the neural spiking patterns, connectivity, and so on that support particular human behaviours. Neuroscience would in turn provide constraints on high-level algorithms, indicating what operations can be performed by neurons, how accurate they can be, and how much time is needed. Our approach uses the Neural Engineering Framework (NEF; Eliasmith & Anderson, 2003), a general method for constructing computational models whose components are simulations of spiking neurons. The NEF provides a method for defining how values can be represented in a distributed manner across a set of neurons. Most crucially, it also allows us to determine how groups of neurons can be synaptically connected such that they will compute desired functions. This allows us to take a cognitive theory expressed in terms of numerical values and transformations on those values and create a detailed neural model. The Tower of Hanoi The task considered here is the Tower of Hanoi, which has a rich history in cognitive science as a problem solving task (e.g. Simon, 1975). Many symbolic (non-neural) cognitive models exist which match expert human behaviour well (e.g. Altmann & Trafton, 2002). The task involves three pegs and a fixed number of disks of different sizes with holes in them such that they can be placed on the pegs. Given a starting position, the goal is to move the disks to a goal position, subject to the constraints that only one disk can be moved at a time and a larger disk cannot be placed on top of a smaller disk. Figure 1 shows the optimal series of steps needed to solve the four-disk Tower of Hanoi when all disks start on one peg and must all be moved onto a different peg. There are many algorithms that could be used to produce this series of steps, and cognitive research on this task involves determining which algorithm(s) people are using by examining factors such as the time taken between steps and the types of errors produced. Anderson, Kushmerick, and Lebiere (1993) provide a variety of measures of human performance on this task, and Figure 1 compares their empirical data to our model performance in terms of the time delay between movements (only conditions where no mistakes were made are considered here).