Neuronal Representation of Numerosity Zero in the Primate Parieto-Frontal Number Network

The calculating brain

The human brain possesses neural networks and mechanisms enabling the representation of numbers, basic arithmetic operations, and mathematical reasoning. Without the ability to represent numerical quantity and perform calculations, our scientifically and technically advanced culture would not exist. However, the origins of numerical abilities are grounded in an intuitive understanding of quantity deeply rooted in biology. Nevertheless, more advanced symbolic arithmetic skills require a cultural background with formal mathematical education. In the past two decades, cognitive neuroscience has seen significant progress in understanding the workings of the calculating brain through various methods and model systems. This review begins by exploring the mental and neuronal representations of nonsymbolic numerical quantity and then progresses to symbolic representations acquired in childhood. During arithmetic operations (addition, subtraction, multiplication, and division), these representations are processed and transformed according to arithmetic rules and principles, leveraging different mental strategies and types of arithmetic knowledge that can be dissociated in the brain. Although it was once believed that number processing and calculation originated from the language faculty, it is now evident that mathematical and linguistic abilities are primarily processed independently in the brain. Understanding how the healthy brain processes numerical information is crucial for gaining insights into debilitating numerical disorders, including acquired conditions like acalculia and learning-related calculation disorders such as developmental dyscalculia.

Single-neuron representation of nonsymbolic and symbolic number zero in the human medial temporal lobe

The number zero holds a special status among numbers, indispensable for developing a comprehensive number theory.1,2,3,4 Despite its importance in mathematics, the neuronal foundation of zero in the human brain is unknown. We conducted single-neuron recordings in neurosurgical patients5,6,7 while they made judgments involving nonsymbolic number representations (dot numerosity), including the empty set, and symbolic numbers (Arabic numerals), including numeral zero. Neurons showed responsiveness to either the empty set or numeral zero, but not both. Neuronal activity to zero in both nonsymbolic and symbolic formats exhibited a numerical distance effect, indicating that zero representations are integrated together with countable numerosities and positive integers at the low end of the number line.8,9 A boundary in neuronal coding existed between the nonsymbolic empty set and small numerosities, correlating with the relative difficulty in discriminating numerosity zero behaviorally. Conversely, no such boundary was found for symbolic zero activity, suggesting that symbolic representations integrate zero with other numerals along the number line, reconciling its outlier role. The status of zero as a special nonsymbolic numerical quantity is reflected in the activity of neurons in the human brain, which seems to serve as a scaffold for more advanced representations of zero as a symbolic number.

Symbolic and non-symbolic representations of numerical zero in the human brain

Representing the quantity zero as a symbolic concept is considered a unique achievement of abstract human thought.1,2 To conceptualize zero, one must abstract away from the (absence of) sensory evidence to construct a representation of numerical absence: creating "something" out of "nothing."2,3,4 Previous investigations of the neural representation of natural numbers reveal distinct numerosity-selective neural populations that overlap in their tuning curves with adjacent numerosities.5,6 Importantly, a component of this neural code is thought to be invariant across non-symbolic and symbolic numerical formats.7,8,9,10,11 Although behavioral evidence indicates that zero occupies a place at the beginning of this mental number line,12,13,14 in humans zero is also associated with unique behavioral and developmental profiles compared to natural numbers,4,15,16,17 suggestive of a distinct neural basis for zero. We characterized the neural representation of zero in the human brain by employing two qualitatively different numerical tasks18,19 in concert with magnetoencephalography (MEG) recordings. We assay both neural representations of non-symbolic numerosities (dot patterns), including zero (empty sets), and symbolic numerals, including symbolic zero. Our results reveal that neural representations of zero are situated along a graded neural number line shared with other natural numbers. Notably, symbolic representations of zero generalized to predict non-symbolic empty sets. We go on to localize abstract representations of numerical zero to posterior association cortex, extending the purview of parietal cortex in human numerical cognition to encompass representations of zero.10,20.

Conscious Experience of Stimulus Presence and Absence Is Actively Encoded by Neurons in the Crow Brain

The emergence of consciousness from brain activity constitutes one of the great riddles in biology. It is commonly assumed that only the conscious perception of the presence of a stimulus elicits neuronal activation to signify a "neural correlate of consciousness," whereas the subjective experience of the absence of a stimulus is associated with a neuronal resting state. Here, we demonstrate that the two subjective states "stimulus present" and "stimulus absent" are represented by two specialized neuron populations in crows, corvid birds. We recorded single-neuron activity from the nidopallium caudolaterale of crows trained to report the presence or absence of images presented near the visual threshold. Because of the task design, neuronal activity tracking the conscious "present" versus "absent" percept was dissociated from that involved in planning a motor response. Distinct neuron populations signaled the subjective percepts of "present" and "absent" by increases in activation. The response selectivity of these two neuron populations was similar in strength and time course. This suggests a balanced code for subjective "presence" versus "absence" experiences, which might be beneficial when both conscious states need to be maintained active in the service of goal-directed behavior.

Monkey Prefrontal Cortex Learns to Minimize Sequence Prediction Error

In this study, we develop a novel recurrent neural network (RNN) model of pre-frontal cortex that predicts sensory inputs, actions, and outcomes at the next time step. Synaptic weights in the model are adjusted to minimize sequence prediction error, adapting a deep learning rule similar to those of large language models. The model, called Sequence Prediction Error Learning (SPEL), is a simple RNN that predicts world state at the next time step, but that differs from standard RNNs by using its own prediction errors from the previous state predictions as inputs to the hidden units of the network. We show that the time course of sequence prediction errors generated by the model closely matched the activity time courses of populations of neurons in macaque prefrontal cortex. Hidden units in the model responded to combinations of task variables and exhibited sensitivity to changing stimulus probability in ways that closely resembled monkey prefrontal neurons. Moreover, the model generated prolonged response times to infrequent, unexpected events as did monkeys. The results suggest that prefrontal cortex may generate internal models of the temporal structure of the world even during tasks that do not explicitly depend on temporal expectation, using a sequence prediction error minimization learning rule to do so. As such, the SPEL model provides a unified, general-purpose theoretical framework for modeling the lateral prefrontal cortex.

Sequential neuronal processing of number values, abstract decision, and action in the primate prefrontal cortex

Decision-making requires processing of sensory information, comparing the gathered evidence to make a judgment, and performing the action to communicate it. How neuronal representations transform during this cascade of representations remains a matter of debate. Here, we studied the succession of neuronal representations in the primate prefrontal cortex (PFC). We trained monkeys to judge whether a pair of sequentially presented displays had the same number of items. We used a combination of single neuron and population-level analyses and discovered a sequential transformation of represented information with trial progression. While numerical values were initially represented with high precision and in conjunction with detailed information such as order, the decision was encoded in a low-dimensional subspace of neural activity. This decision encoding was invariant to both retrospective numerical values and prospective motor plans, representing only the binary judgment of "same number" versus "different number," thus facilitating the generalization of decisions to novel number pairs. We conclude that this transformation of neuronal codes within the prefrontal cortex supports cognitive flexibility and generalizability of decisions to new conditions.

Cortical quantity representations of visual numerosity and timing overlap increasingly into superior cortices but remain distinct

Many sensory brain areas are organized as topographic maps where neural response preferences change gradually across the cortical surface. Within association cortices, 7-Tesla fMRI and neural model-based analyses have also revealed many topographic maps for quantities like numerosity and event timing, often in similar locations. Numerical and temporal quantity estimations also show behavioral similarities and even interactions. For example, the duration of high-numerosity displays is perceived as longer than that of low-numerosity displays. Such interactions are often ascribed to a generalized magnitude system with shared neural responses across quantities. Anterior quantity responses are more closely linked to behavior. Here, we investigate whether common quantity representations hierarchically emerge by asking whether numerosity and timing maps become increasingly closely related in their overlap, response preferences, and topography. While the earliest quantity maps do not overlap, more superior maps overlap increasingly. In these overlapping areas, some intraparietal maps have consistently correlated numerosity and timing preferences, and some maps have consistent angles between the topographic progressions of numerosity and timing preferences. However, neither of these relationships increases hierarchically like the amount of overlap does. Therefore, responses to different quantities are initially derived separately, then progressively brought together, without generally becoming a common representation. Bringing together distinct responses to different quantities may underlie behavioral interactions and allow shared access to comparison and action planning systems.

Convergent Circuit Computation for Categorization in the Brains of Primates and Songbirds

Categorization is crucial for behavioral flexibility because it enables animals to group stimuli into meaningful classes that can easily be generalized to new circumstances. A most abstract quantitative category is set size, the number of elements in a set. This review explores how categorical number representations are realized by the operations of excitatory and inhibitory neurons in associative telencephalic microcircuits in primates and songbirds. Despite the independent evolution of the primate prefrontal cortex and the avian nidopallium caudolaterale, the neuronal computations of these associative pallial circuits show surprising correspondence. Comparing cellular functions in distantly related taxa can inform about the evolutionary principles of circuit computations for cognition in distinctly but convergently realized brain structures.

Distinct neuronal representation of small and large numbers in the human medial temporal lobe

Whether small numerical quantities are represented by a special subitizing system that is distinct from a large-number estimation system has been debated for over a century. Here we show that two separate neural mechanisms underlie the representation of small and large numbers. We performed single neuron recordings in the medial temporal lobe of neurosurgical patients judging numbers. We found a boundary in neuronal coding around number 4 that correlates with the behavioural transition from subitizing to estimation. In the subitizing range, neurons showed superior tuning selectivity accompanied by suppression effects suggestive of surround inhibition as a selectivity-increasing mechanism. In contrast, tuning selectivity decreased with increasing numbers beyond 4, characterizing a ratio-dependent number estimation system. The two systems with the coding boundary separating them were also indicated using decoding and clustering analyses. The identified small-number subitizing system could be linked to attention and working memory that show comparable capacity limitations.

Computational mechanisms underlying motivation to earn symbolic reinforcers

Reinforcement learning (RL) is a theoretical framework that describes how agents learn to select options that maximize rewards and minimize punishments over time. We often make choices, however, to obtain symbolic reinforcers (e.g. money, points) that can later be exchanged for primary reinforcers (e.g. food, drink). Although symbolic reinforcers are motivating, little is understood about the neural or computational mechanisms underlying the motivation to earn them. In the present study, we examined how monkeys learn to make choices that maximize fluid rewards through reinforcement with tokens. The question addressed here is how the value of a state, which is a function of multiple task features (e.g. current number of accumulated tokens, choice options, task epoch, trials since last delivery of primary reinforcer, etc.), drives value and affects motivation. We constructed a Markov decision process model that computes the value of task states given task features to capture the motivational state of the animal. Fixation times, choice reaction times, and abort frequency were all significantly related to values of task states during the tokens task (n=5 monkeys). Furthermore, the model makes predictions for how neural responses could change on a moment-by-moment basis relative to changes in state value. Together, this task and model allow us to capture learning and behavior related to symbolic reinforcement.

Categorical representation of abstract spatial magnitudes in the executive telencephalon of crows

The ability to group abstract continuous magnitudes into meaningful categories is cognitively demanding but key to intelligent behavior. To explore its neuronal mechanisms, we trained carrion crows to categorize lines of variable lengths into arbitrary "short" and "long" categories. Single-neuron activity in the nidopallium caudolaterale (NCL) of behaving crows reflected the learned length categories of visual stimuli. The length categories could be reliably decoded from neuronal population activity to predict the crows' conceptual decisions. NCL activity changed with learning when a crow was retrained with the same stimuli assigned to more categories with new boundaries ("short", "medium," and "long"). Categorical neuronal representations emerged dynamically so that sensory length information at the beginning of the trial was transformed into behaviorally relevant categorical representations shortly before the crows' decision making. Our data show malleable categorization capabilities for abstract spatial magnitudes mediated by the flexible networks of the crow NCL.

Shared Neural Activity But Distinct Neural Dynamics for Cognitive Control in Monkey Prefrontal and Parietal Cortex

To better understand how prefrontal networks mediate forms of cognitive control disrupted in schizophrenia, we translated a variant of the AX continuous performance task that measures specific deficits in the human disease to 2 male monkeys and recorded neurons in PFC and parietal cortex during task performance. In the task, contextual information instructed by cue stimuli determines the response required to a subsequent probe stimulus. We found parietal neurons encoding the behavioral context instructed by cues that exhibited nearly identical activity to their prefrontal counterparts (Blackman et al., 2016). This neural population switched their preference for stimuli over the course of the trial depending on whether the stimuli signaled the need to engage cognitive control to override a prepotent response. Cues evoked visual responses that appeared in parietal neurons first, whereas population activity encoding contextual information instructed by cues was stronger and more persistent in PFC. Increasing cognitive control demand biased the representation of contextual information toward the PFC and augmented the temporal correlation of task-defined information encoded by neurons in the two areas. Oscillatory dynamics in local field potentials differed between cortical areas and carried as much information about task conditions as spike rates. We found that, at the single-neuron level, patterns of activity evoked by the task were nearly identical between the two cortical areas. Nonetheless, distinct population dynamics in PFC and parietal cortex were evident. suggesting differential contributions to cognitive control.SIGNIFICANCE STATEMENT We recorded neural activity in PFC and parietal cortex of monkeys performing a task that measures cognitive control deficits in schizophrenia. This allowed us to characterize computations performed by neurons in the two areas to support forms of cognitive control disrupted in the disease. Subpopulations of neurons in the two areas exhibited parallel modulations in firing rate; and as a result, all patterns of task-evoked activity were distributed between PFC and parietal cortex. This included the presence in both cortical areas of neurons reflecting proactive and reactive cognitive control dissociated from stimuli or responses in the task. However, differences in the timing, strength, synchrony, and correlation of information encoded by neural activity were evident, indicating differential contributions to cognitive control.

Number selective sensorimotor neurons in the crow translate perceived numerosity into number of actions

Translating a perceived number into a matching number of self-generated actions is a hallmark of numerical reasoning in humans and animals alike. To explore this sensorimotor transformation, we trained crows to judge numerical values in displays and to flexibly plan and perform a matching number of pecks. We report number selective sensorimotor neurons in the crow telencephalon that signaled the impending number of self-generated actions. Neuronal population activity during the sensorimotor transformation period predicted whether the crows mistakenly planned fewer or more pecks than instructed. During sensorimotor transformation, both a static neuronal code characterized by persistently number-selective neurons and a dynamic code originating from neurons carrying rapidly changing numerical information emerged. The findings indicate there are distinct functions of abstract neuronal codes supporting the sensorimotor number system.

In search for consciousness in animals: Using working memory and voluntary attention as behavioral indicators

Whether animals have subjective experiences about the content of their sensory input, i.e., whether they are aware of stimuli, is a notoriously difficult question to answer. If consciousness is present in animals, it must share fundamental characteristics with human awareness. Working memory and voluntary/endogenous attention are suggested as diagnostic features of conscious awareness. Behavioral evidence shows clear signatures of both working memory and voluntary attention as minimal criterium for sensory consciousness in mammals and birds. In contrast, reptiles and amphibians show no sign of either working memory or volitional attention. Surprisingly, some species of teleost fishes exhibit elementary working memory and voluntary attention effects suggestive of possibly rudimentary forms of subjective experience. With the potential exception of honeybees, evidence for conscious processing is lacking in invertebrates. These findings suggest that consciousness is not ubiquitous in the animal kingdom but also not exclusive to humans. The phylogenetic gap between animal taxa argues that evolution does not rely on specific neural substrates to endow distantly related species with basic forms of consciousness.

Does quantity matter to a stingless bee?

Quantitative information is omnipresent in the world and a wide range of species has been shown to use quantities to optimize their decisions. While most studies have focused on vertebrates, a growing body of research demonstrates that also insects such as honeybees possess basic quantitative abilities that might aid them in finding profitable flower patches. However, it remains unclear if for insects, quantity is a salient feature relative to other stimulus dimensions, or if it is only used as a "last resort" strategy in case other stimulus dimensions are inconclusive. Here, we tested the stingless bee Trigona fuscipennis, a species representative of a vastly understudied group of tropical pollinators, in a quantity discrimination task. In four experiments, we trained wild, free-flying bees on stimuli that depicted either one or four elements. Subsequently, bees were confronted with a choice between stimuli that matched the training stimulus either in terms of quantity or another stimulus dimension. We found that bees were able to discriminate between the two quantities, but performance differed depending on which quantity was rewarded. Furthermore, quantity was more salient than was shape. However, quantity did not measurably influence the bees' decisions when contrasted with color or surface area. Our results demonstrate that just as honeybees, small-brained stingless bees also possess basic quantitative abilities. Moreover, invertebrate pollinators seem to utilize quantity not only as "last resort" but as a salient stimulus dimension. Our study contributes to the growing body of knowledge on quantitative cognition in invertebrate species and adds to our understanding of the evolution of numerical cognition.

Young domestic chicks spontaneously represent the absence of objects

Absence is a notion that is usually captured by language-related concepts like zero or negation. Whether nonlinguistic creatures encode similar thoughts is an open question, as everyday behavior marked by absence (of food, of social partners) can be explained solely by expecting presence somewhere else. We investigated 8-day-old chicks’ looking behavior in response to events violating expectations about the presence or absence of an object. We found different behavioral responses to violations of presence and absence, suggesting distinct underlying mechanisms. Importantly, chicks displayed an avian signature of novelty detection to violations of absence, namely a sex-dependent left-eye-bias. Follow-up experiments excluded accounts that would explain this bias by perceptual mismatch or by representing the object at different locations. These results suggest that the ability to spontaneously form representations about the absence of objects likely belongs to the initial cognitive repertoire of vertebrate species.

Numerosity tuning in human association cortices and local image contrast representations in early visual cortex

Human early visual cortex response amplitudes monotonically increase with numerosity (object number), regardless of object size and spacing. However, numerosity is typically considered a high-level visual or cognitive feature, while early visual responses follow image contrast in the spatial frequency domain. We find that, at fixed contrast, aggregate Fourier power (at all orientations and spatial frequencies) follows numerosity closely but nonlinearly with little effect of object size, spacing or shape. This would allow straightforward numerosity estimation from spatial frequency domain image representations. Using 7T fMRI, we show monotonic responses originate in primary visual cortex (V1) at the stimulus's retinotopic location. Responses here and in neural network models follow aggregate Fourier power more closely than numerosity. Truly numerosity tuned responses emerge after lateral occipital cortex and are independent of retinotopic location. We propose numerosity's straightforward perception and neural responses may result from the pervasive spatial frequency analyses of early visual processing.

Sometimes nothing is simply nothing: Automatic processing of empty sets

Previous work using the numerical comparison task has shown that an empty set, the nonsymbolic manifestation of zero, can be represented as the smallest quantity of the numerical magnitude system. In this study, we examined whether an empty set can be represented as such under conditions of automatic processing in which deliberate processing of stimuli magnitudes is not required by the task. In Experiment 1, participants performed physical and numerical comparisons of empty sets (i.e., empty frames) and of other numerosities presented as framed arrays of 1 to 9 dots. The physical sizes of the frames varied within pairs. Both tasks revealed a size congruity effect (SCE) for comparisons of non-empty sets. In contrast, comparisons to empty sets produced an inverted SCE in the physical comparison task, whereas no SCE was found for comparisons to empty sets in the numerical comparison task. In Experiment 2, participants performed an area comparison task using the same stimuli as Experiment 1 to examine the effect of visual cues on the automatic processing of empty sets. The results replicated the findings of the physical comparison task in Experiment 1. Taken together, our findings indicate that empty sets are not perceived as "zero," but rather as "nothing," when processed automatically. Hence, the perceptual dominance of empty sets seems to play a more important role under conditions of automatic processing, making it harder to abstract the numerical meaning of zero from empty sets.

The approximate number system represents rational numbers: The special case of an empty set

We agree with Clarke and Beck that the approximate number system represents rational numbers, and we demonstrate our support by highlighting the case of the empty set - the non-symbolic manifestation of zero. It is particularly interesting because of its perceptual and semantic uniqueness, and its exploration reveals fundamental new insights about how numerical information is represented.

Spontaneous representation of numerosity zero in a deep neural network for visual object recognition

Conceiving "nothing" as a numerical value zero is considered a sophisticated numerical capability that humans share with cognitively advanced animals. We demonstrate that representation of zero spontaneously emerges in a deep learning neural network without any number training. As a signature of numerical quantity representation, and similar to real neurons from animals, numerosity zero network units show maximum activity to empty sets and a gradual decrease in activity with increasing countable numerosities. This indicates that the network spontaneously ordered numerosity zero as the smallest numerical value along the number line. Removal of empty-set network units caused specific deficits in the network's judgment of numerosity zero, thus reflecting these units' functional relevance. These findings suggest that processing visual information is sufficient for a visual number sense that includes zero to emerge and explains why cognitively advanced animals with whom we share a nonverbal number system exhibit rudiments of numerosity zero.

The macaque ventral intraparietal area has expanded into three homologue human parietal areas

The macaque ventral intraparietal area (VIP) in the fundus of the intraparietal sulcus has been implicated in a diverse range of sensorimotor and cognitive functions such as motion processing, multisensory integration, processing of head peripersonal space, defensive behavior, and numerosity coding. Here, we exhaustively review macaque VIP function, cytoarchitectonics, and anatomical connectivity and integrate it with human studies that have attempted to identify a potential human VIP homologue. We show that human VIP research has consistently identified three, rather than one, bilateral parietal areas that each appear to subsume some, but not all, of the macaque area's functionality. Available evidence suggests that this human "VIP complex" has evolved as an expansion of the macaque area, but that some precursory specialization within macaque VIP has been previously overlooked. The three human areas are dominated, roughly, by coding the head or self in the environment, visual heading direction, and the peripersonal environment around the head, respectively. A unifying functional principle may be best described as prediction in space and time, linking VIP to state estimation as a key parietal sensorimotor function. VIP's expansive differentiation of head and self-related processing may have been key in the emergence of human bodily self-consciousness.

Feature-based attention processes in primate prefrontal cortex do not rely on feature similarity

Feature-based attention enables privileged processing of specific visual properties. During feature-based attention, neurons in visual cortices show "gain modulation" by enhancing neuronal responses to the features of attended stimuli due to top-down signals originating from prefrontal cortex (PFC). Attentional modulation in visual cortices requires "feature similarity:" neurons only increase their responses when the attended feature variable and the neurons' preferred feature coincide. However, whether gain modulation based on feature similarity is a general attentional mechanism is currently unknown. To address this issue, we record single-unit activity from PFC of macaques trained to switch attention between two conjunctive feature parameters. We find that PFC neurons experience gain modulation in response to attentional demands. However, this attentional gain modulation in PFC is independent of the feature-tuning preferences of neurons. These findings suggest that feature similarity is not a general mechanism in feature-based attention throughout the cortical processing hierarchy.

Development of Preschoolers' Understanding of Zero

While knowledge on the development of understanding positive integers is rapidly growing, the development of understanding zero remains not well-understood. Here, we test several components of preschoolers’ understanding of zero: Whether they can use empty sets in numerical tasks (as measured with comparison, addition, and subtraction tasks); whether they can use empty sets soon after they understand the cardinality principle (cardinality-principle knowledge is measured with the give-N task); whether they know what the word “zero” refers to (tested in all tasks in this study); and whether they categorize zero as a number (as measured with the smallest-number and is-it-a-number tasks). The results show that preschoolers can handle empty sets in numerical tasks as soon as they can handle positive numbers and as soon as, or even earlier than, they understand the cardinality principle. Some also know that these sets are labeled as “zero.” However, preschoolers are unsure whether zero is a number. These results identify three components of knowledge about zero: operational knowledge, linguistic knowledge, and meta-knowledge. To account for these results, we propose that preschoolers may understand numbers as the properties of items or objects in a set. In this view, zero is not regarded as a number because an empty set does not include any items, and missing items cannot have any properties, therefore, they cannot have the number property either. This model can explain why zero is handled correctly in numerical tasks even though it is not regarded as a number.

Behavioral and Neuronal Representation of Numerosity Zero in the Crow

Different species of animals can discriminate numerosity, the countable number of objects in a set. The representations of countable numerosities have been deciphered down to the level of single neurons. However, despite its importance for human number theory, a special numerical quantity, the empty set (numerosity zero), has remained largely unexplored. We explored the behavioral and neuronal representation of the empty set in carrion crows. Crows were trained to discriminate small numerosities including the empty set. Performance data showed a numerical distance effect for the empty set in one crow, suggesting that the empty set and countable numerosities are represented along the crows' "mental number line." Single-cell recordings in the endbrain region nidopallium caudolaterale (NCL) showed a considerable proportion of NCL neurons tuned to the preferred numerosity zero. As evidenced by neuronal distance and size effects, NCL neurons integrated the empty set in the neural number line. A subsequent neuronal population analysis using a statistical classifier approach showed that the neuronal numerical representations were predictive of the crows' success in the task. These behavioral and neuronal data suggests that the conception of the empty set as a cognitive precursor of a zero-like number concept is not an exclusive property of the cerebral cortex of primates. Zero as a quantitative category cannot only be implemented in the layered neocortex of primates, but also in the anatomically distinct endbrain circuitries of birds that evolved based on convergent evolution.SIGNIFICANCE STATEMENT The conception of "nothing" as number "zero" is celebrated as one of the greatest achievements in mathematics. To explore whether precursors of zero-like concepts can be found in vertebrates with a cerebrum that anatomically differs starkly from our primate brain, we investigated this in carrion crows. We show that crows can grasp the empty set as a null numerical quantity that is mentally represented next to number one. Moreover, we show that single neurons in an associative avian cerebral region specifically respond to the empty set and show the same physiological characteristics as for countable quantities. This suggests that zero as a quantitative category can also be implemented in the anatomically distinct endbrain circuitries of birds that evolved based on convergent evolution.

The Evolutionary History of Brains for Numbers

Humans and other animals share a number sense', an intuitive understanding of countable quantities. Having evolved independent from one another for hundreds of millions of years, the brains of these diverse species, including monkeys, crows, zebrafishes, bees, and squids, differ radically. However, in all vertebrates investigated, the pallium of the telencephalon has been implicated in number processing. This suggests that properties of the telencephalon make it ideally suited to host number representations that evolved by convergent evolution as a result of common selection pressures. In addition, promising candidate regions in the brains of invertebrates, such as insects, spiders, and cephalopods, can be identified, opening the possibility of even deeper commonalities for number sense.

Nonsymbolic and symbolic representations of null numerosity

Previous research has shown that null numerosity can be processed as a numerical entity that is represented together with non-null numerosities on the same magnitude system. The present study examined which conditions enable perceiving nonsymbolic (i.e., an empty set) and symbolic (i.e., 0) representations of null numerosity as a numerical entity, using distance and end effects. In Experiment 1, participants performed magnitude comparisons of notation homogeneous pairs (both numerosities appeared in nonsymbolic or symbolic format), as well as heterogeneous pairs (a nonsymbolic numerosity versus a symbolic one). Comparisons to 0 resulted in faster responses and an attenuated distance effect in all conditions, whereas comparisons to an empty set produced such effects only in the nonsymbolic and symbolic homogeneous conditions. In Experiments 2 and 3, participants performed same/different numerosity judgments with heterogeneous pairs. A distance effect emerged for "different" judgments of 0 and sets of 1 to 9 dots, but not for those with an empty set versus digits 1-9. These findings indicate that perceiving an empty set, but not 0, as a numerical entity is determined by notation homogeneity and task requirements.

Neuroethology of number sense across the animal kingdom

Many species from diverse and often distantly related animal groups (e.g. monkeys, crows, fish and bees) have a sense of number. This means that they can assess the number of items in a set - its 'numerosity'. The brains of these phylogenetically distant species are markedly diverse. This Review examines the fundamentally different types of brains and neural mechanisms that give rise to numerical competence across the animal tree of life. Neural correlates of the number sense so far exist only for specific vertebrate species: the richest data concerning explicit and abstract number representations have been collected from the cerebral cortex of mammals, most notably human and nonhuman primates, but also from the pallium of corvid songbirds, which evolved independently of the mammalian cortex. In contrast, the neural data relating to implicit and reflexive numerical representations in amphibians and fish is limited. The neural basis of a number sense has not been explored in any protostome so far. However, promising candidate regions in the brains of insects, spiders and cephalopods - all of which are known to have number skills - are identified in this Review. A comparative neuroscientific approach will be indispensable for identifying evolutionarily stable neuronal circuits and deciphering codes that give rise to a sense of number across phylogeny.

Non-numerical strategies used by bees to solve numerical cognition tasks

We examined how bees solve a visual discrimination task with stimuli commonly used in numerical cognition studies. Bees performed well on the task, but additional tests showed that they had learned continuous (non-numerical) cues. A network model using biologically plausible visual feature filtering and a simple associative rule was capable of learning the task using only continuous cues inherent in the training stimuli, with no numerical processing. This model was also able to reproduce behaviours that have been considered in other studies indicative of numerical cognition. Our results support the idea that a sense of magnitude may be more primitive and basic than a sense of number. Our findings highlight how problematic inadvertent continuous cues can be for studies of numerical cognition. This remains a deep issue within the field that requires increased vigilance and cleverness from the experimenter. We suggest ways of better assessing numerical cognition in non-speaking animals, including assessing the use of all alternative cues in one test, using cross-modal cues, analysing behavioural responses to detect underlying strategies, and finding the neural substrate.

Distinct neural networks for the volitional control of vocal and manual actions in the monkey homologue of Broca's area

The ventrolateral frontal lobe (Broca's area) of the human brain is crucial in speech production. In macaques, neurons in the ventrolateral prefrontal cortex, the suggested monkey homologue of Broca's area, signal the volitional initiation of vocalizations. We explored whether this brain area became specialized for vocal initiation during primate evolution and trained macaques to alternate between a vocal and manual action in response to arbitrary cues. During task performance, single neurons recorded from the ventrolateral prefrontal cortex and the rostroventral premotor cortex of the inferior frontal cortex predominantly signaled the impending vocal or, to a lesser extent, manual action, but not both. Neuronal activity was specific for volitional action plans and differed during spontaneous movement preparations. This implies that the primate inferior frontal cortex controls the initiation of volitional utterances via a dedicated network of vocal selective neurons that might have been exploited during the evolution of Broca's area.

On the indicators for perceiving empty sets as zero

The question whether human beings process empty sets as zero has received little research attention. In this study, we used the distance and end effects as indicators for treating empty sets as a numerical entity that represents an absence of quantity. In a series of experiments, participants performed a magnitude comparison task. They were presented with empty sets and other numerosities from 1 to 9, presented as dot arrays. We manipulated task instructions relevant to the target (i.e., "choose the target that contains more/less dots" in Experiment 1) or the given numerical range mentioned in the instructions (i.e., 0-9 or 1-9 in Experiment 2) to create conditions in which an empty set would be perceived as the smallest value of the experimental numerical range. The results revealed distance effects for comparisons to empty sets, irrespective of task instructions. In Experiment 3, we manipulated the response mode. Two groups of participants responded to target location, one group with a key-press and the other vocally, while the third group responded vocally to target color. The results revealed distance effects for comparisons to empty sets only when responding to target location, regardless of the response mode, indicating that spatial features should be primed in order to perceive an empty set as a numerical entity. These findings show that perceiving an empty set as nothing or as zero depends on the context in which it is presented.

A sense of number in invertebrates

Non-symbolic numerical abilities are widespread among vertebrates due to their important adaptive value. Moreover, these abilities were considered peculiar of vertebrate species as numerical competence is regarded as cognitively sophisticated. However, recent evidence convincingly showed that this is not the case: invertebrates, with their limited number of neurons, proved able to successfully discriminate different quantities (e.g., of prey), to use the ordinal property of numbers, to solve arithmetic operations as addition and subtraction and even to master the concept of zero numerosity. To date, though, the debate is still open on the presence and the nature of a «sense of number» in invertebrates. Whether this is peculiar for discrete countable quantities (numerosities) or whether this is part of a more general magnitude system dealing with both discrete and continuous quantities, as hypothesized for humans and other vertebrates. Here we reviewed the main studies on numerical abilities of invertebrates, discussing in particular the recent findings supporting the hypothesis of a general mechanism that allows for processing of both discrete (i.e., number) and continuous dimensions (e.g., space).

Spatial Neuronal Integration Supports a Global Representation of Visual Numerosity in Primate Association Cortices

Our sense of number rests on the activity of neurons that are tuned to the number of items and show great invariance across display formats and modalities. Whether numerosity coding becomes abstracted from local spatial representations characteristic of visual input is not known. We mapped the visual receptive fields (RFs) of numerosity-selective neurons in the pFC and ventral intraparietal area in rhesus monkeys. We found numerosity selectivity in pFC and ventral intraparietal neurons irrespective of whether they exhibited an RF and independent of the location of their RFs. RFs were not predictive of the preference of numerosity-selective neurons. Furthermore, the presence and location of RFs had no impact on tuning width and quality of the numerosity-selective neurons. These findings show that neurons in frontal and parietal cortices integrate abstract visuospatial stimuli to give rise to global and spatially released number representations as required for number perception.

The neurobiology of innate, volitional and learned vocalizations in mammals and birds

Vocalization is an ancient vertebrate trait essential to many forms of communication, ranging from courtship calls to free verse. Vocalizations may be entirely innate and evoked by sexual cues or emotional state, as with many types of calls made in primates, rodents and birds; volitional, as with innate calls that, following extensive training, can be evoked by arbitrary sensory cues in non-human primates and corvid songbirds; or learned, acoustically flexible and complex, as with human speech and the courtship songs of oscine songbirds. This review compares and contrasts the neural mechanisms underlying innate, volitional and learned vocalizations, with an emphasis on functional studies in primates, rodents and songbirds. This comparison reveals both highly conserved and convergent mechanisms of vocal production in these different groups, despite their often vast phylogenetic separation. This similarity of central mechanisms for different forms of vocal production presents experimentalists with useful avenues for gaining detailed mechanistic insight into how vocalizations are employed for social and sexual signalling, and how they can be modified through experience to yield new vocal repertoires customized to the individual's social group. This article is part of the theme issue 'What can animal communication teach us about human language?'

Neural constraints on human number concepts

True counting and arithmetic abilities are unique to humans and are inextricably linked to symbolic competence. However, our unprecedented numerical skills are deeply rooted in our neuronal heritage as primates and vertebrates. In this article, I argue that numerical competence in humans is the result of three neural constraints. First, I propose that the neuronal mechanisms of quantity estimation are part of our evolutionary heritage and can be witnessed across primate and vertebrate phylogeny. Second, I suggest that a basic understanding of number, what numerical quantity means, is innately wired into the brain and gives rise to an intuitive number sense, or number instinct. Third and finally, I argue that symbolic counting and arithmetic in humans is rooted in an evolutionarily and ontogenetically primeval neural system for non-symbolic number representations. These three neural constraints jointly determine the basic processing of number concepts in the human mind.

Absence perception and the philosophy of zero

Zero provides a challenge for philosophers of mathematics with realist inclinations. On the one hand it is a bona fide cardinal number, yet on the other it is linked to ideas of nothingness and non-being. This paper provides an analysis of the epistemology and metaphysics of zero. We develop several constraints and then argue that a satisfactory account of zero can be obtained by integrating (1) an account of numbers as properties of collections, (2) work on the philosophy of absences, and (3) recent work in numerical cognition and ontogenetic studies.

Working memory representation of empty sets in the primate parietal and prefrontal cortices

For the brain, representing empty sets as a precursor to zero is a challenge because it requires the active coding of a quantitative category that, by definition, contains no items. Recent neurophysiological recordings show that empty sets are distinctively encoded by neurons in the primate ventral intraparietal area (VIP) and the prefrontal cortex (PFC). However, how empty sets are represented in working memory is unknown. We simultaneously recorded from VIP and PFC while rhesus monkeys performed a delayed numerosity matching task that required the maintenance of numerosities in memory for a brief period. Countable numerosities (1-4) and empty sets ('numerosity 0') were included as stimuli. Single neurons in PFC, and to a lesser extent neurons in VIP, actively encoded empty sets during the delay period. In both cortical areas, empty sets were progressively differentiated from countable numerosities with time during the ongoing trial. Moreover, the tuning of neuron populations in VIP and PFC shifted dynamically towards empty sets so that they became increasingly overrepresented in working memory. Compared to VIP, the prefrontal representation of empty sets was more stable in time and more independent of low level visual features. Moreover, PFC activity correlated better with behavioral performance in empty set trials. These findings suggest that the representation of null quantity in working memory relies more on prefrontal and less on parietal processing. Overall, our results show that empty sets are dynamically and distinctly represented in working memory.

Computing Value from Quality and Quantity in Human Decision-Making

How organisms learn the value of single stimuli through experience is well described. In many decisions, however, value estimates are computed “on the fly” by combining multiple stimulus attributes. The neural basis of this computation is poorly understood. Here we explore a common scenario in which decision-makers must combine information about quality and quantity to determine the best option. Using fMRI, we examined the neural representation of quality, quantity, and their integration into an integrated subjective value signal in humans of both genders. We found that activity within inferior frontal gyrus (IFG) correlated with offer quality, while activity in the intraparietal sulcus (IPS) specifically correlated with offer quantity. Several brain regions, including the anterior cingulate cortex (ACC), were sensitive to an interaction of quality and quantity. However, the ACC was uniquely activated by quality, quantity, and their interaction, suggesting that this region provides a substrate for flexible computation of value from both quality and quantity. Furthermore, ACC signals across subjects correlated with the strength of quality and quantity signals in IFG and IPS, respectively. ACC tracking of subjective value also correlated with choice predictability. Finally, activity in the ACC was elevated for choice trials, suggesting that ACC provides a nexus for the computation of subjective value in multiattribute decision-making.

SIGNIFICANCE STATEMENT Would you prefer three apples or two oranges? Many choices we make each day require us to weigh up the quality and quantity of different outcomes. Using fMRI, we show that option quality is selectively represented in the inferior frontal gyrus, while option quantity correlates with areas of the intraparietal sulcus that have previously been associated with numerical processing. We show that information about the two is integrated into a value signal in the anterior cingulate cortex, and the fidelity of this integration predicts choice predictability. Our results demonstrate how on-the-fly value estimates are computed from multiple attributes in human value-based decision-making.

Numerical ordering of zero in honey bees

Some vertebrates demonstrate complex numerosity concepts-including addition, sequential ordering of numbers, or even the concept of zero-but whether an insect can develop an understanding for such concepts remains unknown. We trained individual honey bees to the numerical concepts of "greater than" or "less than" using stimuli containing one to six elemental features. Bees could subsequently extrapolate the concept of less than to order zero numerosity at the lower end of the numerical continuum. Bees demonstrated an understanding that parallels animals such as the African grey parrot, nonhuman primates, and even preschool children.

Evolution of cognitive and neural solutions enabling numerosity judgements: lessons from primates and corvids

Brains that are capable of representing numerosity, the number of items in a set, have arisen repeatedly and independently in different animal taxa. This review compares the cognitive and physiological mechanisms found in a nonhuman primate, the rhesus macaque, and a corvid songbird, the carrion crow, in order to elucidate the evolutionary adaptations underlying numerical competence. Monkeys and corvids are known for their advanced cognitive competence, despite them both having independently and distinctly evolved endbrains that resulted from a long history of parallel evolution. In both species, numerosity is represented as an analogue magnitude by an approximate number system that obeys the Weber-Fechner Law. In addition, the activity of numerosity-selective neurons in the fronto-parietal association cortex of monkeys and the telencephalic associative area nidopallium caudolaterale of crows mirrors the animals' performance. In both species' brains, neuronal activity is tuned to a preferred numerosity, encodes the numerical value in an approximate fashion, and is best represented on a logarithmic scale. Collectively, the data show an impressive correspondence of the cognitive and neuronal mechanisms for numerosity representations across monkeys and crows. This suggests that remotely related vertebrates with distinctly developed endbrains adopted similar physiological solutions to common computational problems in numerosity processing.This article is part of a discussion meeting issue 'The origins of numerical abilities'.

A random-matrix theory of the number sense

Number sense, a spontaneous ability to process approximate numbers, has been documented in human adults, infants and newborns, and many other animals. Species as distant as monkeys and crows exhibit very similar neurons tuned to specific numerosities. How number sense can emerge in the absence of learning or fine tuning is currently unknown. We introduce a random-matrix theory of self-organized neural states where numbers are coded by vectors of activation across multiple units, and where the vector codes for successive integers are obtained through multiplication by a fixed but random matrix. This cortical implementation of the 'von Mises' algorithm explains many otherwise disconnected observations ranging from neural tuning curves in monkeys to looking times in neonates and cortical numerotopy in adults. The theory clarifies the origin of Weber-Fechner's Law and yields a novel and empirically validated prediction of multi-peak number neurons. Random matrices constitute a novel mechanism for the emergence of brain states coding for quantity.This article is part of a discussion meeting issue 'The origins of numerical abilities'.

Neurons in the Endbrain of Numerically Naive Crows Spontaneously Encode Visual Numerosity

Endowed with an elaborate cerebral cortex, humans and other primates can assess the number of items in a set, or numerosity, from birth on [1] and without being trained [2]. Whether spontaneous numerosity extraction is a unique feat of the mammalian cerebral cortex [3-7] or rather an adaptive property that can be found in differently designed and independently evolved neural substrates, such as the avian enbrain [8], is unknown. To address this question, we recorded single-cell activity from the nidopallium caudolaterale (NCL), a high-level avian association brain area [9-11], of numerically naive crows. We found that a proportion of NCL neurons were spontaneously responsive to numerosity and tuned to the number of items, even though the crows were never trained to assess numerical quantity. Our data show that numerosity-selective neuronal responses are spontaneously present in the distinct endbrains of diverge vertebrate taxa. This seemingly hard-wired property of the avian endbrain to extract numerical quantity explains how birds in the wild, or right after hatching, can exploit numerical cues when making foraging or social decisions. It suggests that endbrain circuitries that evolved based on convergent evolution, such as the avian endbrain, give rise to the same numerosity code.

Number faculty is alive and kicking: On number discriminations and number neurons

Leibovich et al. advocate for a single "sense of magnitude" to which a dedicated faculty for number could allegedly be reduced. This conclusion is unjustified as the authors adopt an unnecessarily narrow definition of "number sense," neglect studies that demonstrate non-symbolic numerosity representation, and furthermore ignore abstract number representations in the brain.

The zero effect: voxel-based lesion symptom mapping of number transcoding errors following stroke

Zero represents a special case in our numerical system because it is not represented on a semantic level. Former research has shown that this can lead to specific impairments when transcoding numerals from dictation to written digits. Even though, number processing is considered to be dominated by the left hemisphere, studies have indicated that both left as well as right hemispheric stroke patients commit errors when transcoding numerals including zeros. Here, for the first time, a large sample of subacute stroke patients (N = 667) was assessed without being preselected based on the location of their lesion, or a specific impairment in transcoding zero. The results show that specific errors in transcoding zeros were common (prevalence = 14.2%) and a voxel-based lesion symptom mapping analysis (n = 153) revealed these to be related to lesions in and around the right putamen. In line with former research, the present study argues that the widespread brain network for number processing also includes subcortical regions, like the putamen with connections to the insular cortex. These play a crucial role in auditory perception as well as attention. If these areas are lesioned, number processing tasks with higher attentional and working memory loads, like transcoding zeros, can be impaired.

Sensory and Working Memory Representations of Small and Large Numerosities in the Crow Endbrain

Neurons in the avian nidopallium caudolaterale (NCL), an endbrain structure that originated independently from the mammalian neocortex, process visual numerosities. To clarify the code for number in this anatomically distinct endbrain area in birds, neuronal responses to a broad range of numerosities were analyzed. We recorded single-neuron activity from the NCL of crows performing a delayed match-to-sample task with visual numerosities as discriminanda. The responses of >20% of randomly selected neurons were modulated significantly by numerosities ranging from one to 30 items. Numerosity-selective neurons showed bell-shaped tuning curves with one of the presented numerosities as preferred numerosity regardless of the physical appearance of the items. The resulting labeled-line code exhibited logarithmic compression obeying the Weber-Fechner law for magnitudes. Comparable proportions of selective neurons were found, not only during stimulus presentation, but also in the delay phase, indicating a dominant role of the NCL in numerical working memory. Both during sensory encoding and memorization of numerosities in working memory, NCL activity predicted the crows' number discrimination performance. These neuronal data reveal striking similarities across vertebrate taxa in their code for number despite convergently evolved and anatomically distinct endbrain structures.

SIGNIFICANCE STATEMENT

Birds are known for their capabilities to process numerical quantity. However, birds lack a six-layered neocortex that enables primates with numerical competence. We aimed to decipher the neuronal code for numerical quantity in the independently and distinctly evolved endbrain of birds. We recorded the activity of neurons in an endbrain association area termed nidopallium caudolaterale (NCL) from crows that assessed and briefly memorized numerosities from one to 30 dots. We report a neuronal code for sensory representation and working memory of numerosities in the crow NCL exhibiting several characteristics that are surprisingly similar to the ones found in primates. Our data suggest a common code for number in two different vertebrate taxa that has evolved based on convergent evolution.