How does the brain code quantity?
Professor Charles Gallistel, Rutgers University, USA
The representation of numerosity must be embedded in a system for representing both discrete and continuous quantities. The most basic question then is the coding question: how does the brain encode a quantity? Neuroscience has nothing to offer in the way of an answer. Computer science does. The code is probably not unary, which rules out analog codes, like rate codes, because it must support the implementation of addition and multiplication over many orders of magnitude. Also analog codes have reading noise; the build up of purely computational error makes dead reckoning impossible. The representation must be able to approximate a huge range of the computable numbers (negative, positive, |n|>> 1 and |n|<<1). Twos-complement fixed point with settable offset (bias) and slope (scale) and a 2-bit to 8-bit integer portion has much to recommend it: 1) it obeys Weber’s Law. 2) It is the most computationally efficient of the known codes (least amount of hardware and least energy use). 3) It is particularly advised for Archimedean computations like dead reckoning, where a great many additions of small quantities may produce a quantity orders of magnitude larger. 4) It converts subtraction to addition. 5) The bit flipping that is the key to this conversion has a natural chemical realisation. 6) Division and multiplication by 2 are particularly simple and known to be psychologically easy and fast. 7) It can produce an exact code for large integers. 8) It solves the exact equality problem.
Numerical assessment in the wild: insights from social carnivores and other mammals
Professor Sarah Benson-Amram, University of Wyoming, USA
Playback experiments have proven to be a useful tool to investigate the extent to which wild animals understand numerical concepts and the factors that play into their decisions to respond to different numbers of vocalising conspecifics. Professor Benson-Amram will review a series of playback experiments conducted with wild social carnivores and other mammals, including African lions, spotted hyenas, and chimpanzees, which demonstrate that these animals can assess the number of conspecifics calling and respond based on numerical advantage. Additionally, she will discuss the key role that individual discrimination and cross-modal recognition can play in the ability of animals to assess the number of conspecifics vocalising nearby. For example, a listener hearing three vocalisations would benefit from being able to assess whether the vocalisations were emitted by the same individual or three different ones, and whether the identity of the callers match individuals they have recently seen in the area. Because the costs and benefits associated with approaching conspecifics change depending on the callers’ age, sex, and relatedness, listeners will likely adjust the level of their behavioural response to playback experiments where the identity of the callers change even when the number of callers is held constant. The listener’s sex, age, social rank and social system will also help determine their behavioural response to varying number of competitors. Finally, Professor Benson-Amram will discuss the implications of these findings for understanding how carnivores and other animals may have evolved a concept of ‘one’.
At the roots of numerical cognition: insights from the day-old domestic chick (Gallus gallus)
Dr Rosa Rugani, University of Padova, Italy
The ability to represent number and to use numerical concepts, such as real numbers, logarithms, and square roots, is a prerogative of a subset of human beings who have received specific mathematical instruction. In the last few decades, however, it has been demonstrated that non-verbal numerical abilities (i.e., those calculations that can be solved in the absence of words) are widespread within the animal Kingdom. To investigate the ontogenetic origins of numerical knowledge Dr Rugani used the domestic chick (Gallus gallus) as animal model. Unlike previous studies on adult animals, this model can be tested very early in life, allowing a precise control of sensory experience.
Dr Rugani will discuss evidence revealing that day-old domestic chicks can: (i) discriminate between different numbers of artificial social companions (i.e., objects they were exposed to soon after hatching); (ii) solve rudimentary arithmetic calculations, such as 1+1+1 vs. 1+1; and (iii) use ordinal information, identifying a target element, (e.g., the 4th), in a series of identical elements, on the basis of its numerical position in the series.
These studies suggest that non-verbal numerical comprehension can be observed in animals in the absence of (or with very reduced) experience, indicating that numerical competences may not have emerged ‘de novo’ in our species together with language, but that they could be based on an evolutionary-ancient precursor system.
The primacy of numerical information
Professor Elizabeth Brannon, University of Pennsylvania, USA
The ability to use numbers is one of the most complex cognitive abilities that humans possess and is often held up as a defining feature of the human mind. Alongside the uniquely human symbolic system for representing number we possess an approximate number system (ANS) that is evolutionarily ancient and developmentally conservative. In this talk Professor Brannon will illustrate the signatures of the ANS with experimental data from human babies and nonhuman primates. She will describe behavioural and neurobiological data that demonstrates how the human and nonhuman primate mind privileges numerical information over other types of quantitative information. She will argue that this numerical privilege implicates the biological importance of number in our evolutionary history.
Numerical abilities in fish
Dr Christian Agrillo, University of Padova, Italy
While there is a well established tradition of studying numerical abilities in mammals and birds, it is only in the last decade that some studies have proposed that teleost fish possess similar capacities. There is substantial evidence showing that fish integrate numerical information and continuous quantities (such as cumulative surface area or convex hull) when assessing which group of fish or objects is larger/smaller. Typically, their performance is more accurate when both pieces of information are simultaneously available, although different fish species were also shown to use pure numerical information when prevented from using continuous quantities. The ability to discriminate small numbers of social companions seems to be already displayed at birth while large number discrimination develops later as a consequence of maturation and experience. The similarities among species (i.e., Gambusia holbrooki, Poecilia reticulata, Danio rerio, Pterophyllum scalare and Xenotoca eiseni) appear greater than the differences, and in general, the numerical capacities of fish partially match those reported in mammals and birds, raising the intriguing idea that our non-symbolic numerical abilities are more ancient than previously thought and date back at least as far as the divergence between fish and land vertebrates. Dr Agrillo will summarise the current state of art in the literature, focusing on three main topics: the relation between discrete (numerical) and continuous quantities, the ontogeny of numerical abilities in fish and the comparison of numerical abilities of fish and other vertebrates.
Neural correlates of the numerical abilities of anurans: neurons that count
Professor Gary Rose, University of Utah, USA
Acoustic communication plays important roles in the reproductive behaviour of anurans (frogs and toads). The acoustic repertoire of most species consists of several call types, but some anurans gradually increase the complexity of their calls during aggressive interactions between males and when approached by females. Observations of natural behaviour, as well as experimental studies, have revealed the numerical abilities of anurans in their acoustic communication. In particular, anurans are able to discern the number of properly timed pulses (notes) in their calls. The temporal intervals between successive pulses provide information about species identity and call type. A neural correlate of this numerical ability is evident in the responses of neurons that show ‘tuning’ for mid to fast pulse rates. These ‘interval-counting’ neurons respond only after at least a threshold number of pulses have occurred with the correct timing. A single interpulse interval that is 2-3 times the optimal value can reset this interval-counting process. Whole-cell recordings of the membrane potentials of midbrain neurons, in vivo, have revealed that complex interplay between activity-dependent excitation and inhibition contributes to this counting process. Single pulses primarily elicit inhibition. As additional pulses are presented with optimal intervals, cells are progressively depolarised and spike after a threshold number of intervals have occurred. Similarly, pulses that are presented at long intervals (slow rates) elicit primarily inhibition. As interpulse intervals are shortened, however, depolarisation progressively increases. The mechanisms that underlie this apparent shift in the balance of excitation and inhibition during a pulse sequence are under investigation.
Professor Lars Chittka, Queen Mary University of London, UK
When counting-like abilities were first described in the honeybee in the mid 1990s, many scholars were sceptical, but such capacities have since been confirmed in a number of paradigms and also in other insect species, though curiously not in a solitary bee species in a natural foraging task. Counter to the intuitive notion that counting is a cognitively advanced ability, neural network analyses indicate that they can be mediated by very small neural circuits, and we should therefore perhaps not be surprised that insects and other small brained animals such as some small fish exhibit such abilities. One outstanding question is how bees actually acquire numerical information. Recent work on the question of whether bees can see ‘at a glance’ indicates that bees must acquire spatial detail by sequential scanning rather than parallel processing. This is confirmed for a numerosity task in the bumblebee.
Comparative cognition of space and number: the case of the mental number line
Professor Giorgio Vallortigara, University of Trento, Italy
Evidence will be discussed about encoding of geometry and forming associations between space and numbers in non-human animals. A variety of vertebrate species are able to reorient in a rectangular environment in accord with its metrical and sense relations, i.e. using simple Euclidian geometry. There seems to be a primacy of geometric over non-geometric information and, possibly, innate encoding of the sense of direction. Moreover, the hippocampal formation plays a key role in geometry navigation in mammals, birds and fish. Although some invertebrate species show similar behaviours, it is unclear whether the underlying mechanisms are shared. A disposition to associate numerical magnitudes onto a left-to-right-oriented mental number line appears to exist independently of cultural factors, and can be observed in animals with very little numerical experience, such as three-day old chicks. This evidence supports a nativistic foundation of such orientation. Preliminary evidence suggests that the same is observed in human neonates and in zebrafish.